A COMPARATIVE STUDY OF LOWER BOUND
BEARING CAPACITY SOLUTIONS
A Thesis Submitted
in Partial Fulfilment of the Requirements
for the Degree of
MASTER OF TECHNOLOGY
By
SANJAY KUAMR SRIVASTAVA
to the
DEPARTMENT OF CIVIL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY KANPUR
May, 1993
-3 DEC 1993
ce;vtpal
• ^ '• i-
v„ aIj fivsE
C^ - 199 ^ -h'SRl ' c ovo
CERTIFICATE
It is certified that the work contained in the thesis entitled A COMPARATIVE
STUDY OF LOWER BOUND BEARING CAPACITY SOLUTIONS by
SANJAY KUMAR SRIVASTAVA has been carried out under my supervision and that
this work has not been submitted elsewhere for a degree.
(XV
P. K. Basudhar
Professor
Dept, of Civil Engineering
Indian Institute of Technology
Kanpur-208016, INDIA
May, 1993
ACKNOWLEDGEMENTS
To the utmost depth of my heart, I cannot visualize the epitomizing of my thesis
work without the help of some of my nears and dears.
First and the foremost, I would like to express my deep sense of gratitude and appre-
ciation to Professor P.K.Basudhar for his inspiring guidance and arduous supervision
which were instrumental in the completion of this work. He provided me with neces-
sary manoeuvrability and freedom to work, a feature of his guidance, albeit keeping
a watchful eye on the progress. I sincerely cherish his words of encouragements and
counsel.
A special words of appreciation is due to the faculties of civil engineering department
under whose able guidance I could gain a knowledge of this field.
I owe heartfelt gratitude towards all of my caring friends, without mentioning their
names, who have in all possible ways extended their help as and when needed.
SAN JAY KUMAR SRIVASTAVA
ABSTRACT
Sanjay Kumar Srivastava
Roll No. 9110328
Department, of Civil Engineering
Indian Institute of Technology, Kanpur-208 016
India
A COMPARATIVE STUDY OF LOWER BOUND
BEARING CAPACITY SOLUTIONS
The primary object of this thesis is to compare the optimal lower bound bearing capacity
solutions obtained by using the modified Lysmer’s approach (Lysmer-Basudhar) with other
solutions available in literature to asses its capability vis-a-vis other methods based on linear
as well as non-linear programming techniques for isolation of the optimal stress field. This
has been done with reference to bearing capacity of strip footings resting on the surface of
a homogeneous soil deposit. Then by extending the Lysmer-Basudhar approach to bearing
capacity of two layered soil deposits, similar study has been carried out and the obtained
results have been compared with experimental observations and solutions based on method
of characteristics and non-linear programming technique. The comparisons show that all
the methods predict the bearing capacity reasonably well. For stratified deposits, Lysmer-
Basudhar approach predicts values which are very close to experimental results.
CONTENTS
CHAPTER 1 INTRODUCTION 1
1.1 General 1
1.2 Brief Review of Literature 1
1.3 Motivation of the Work 19
1.4 Scope and Organization 21
CHAPTER 2 GENERAL METHOD OF ANALYSIS 23
2.1 General 23
2.2 Element Equilibrium 24
2.3 Interface Equilibrium 28
2.4 External Boundary Conditions 29
2.5 No-yield Conditions 29
2.6 Objective Function, Design Variables, Design Restrictions 31
and Reduction of Design Variables
2.7 Mathematical Programming Problem 35
CHAPTER 3 LOWER BOUND BEARING CAPACITY 37
OF SURFACE STRIP FOOTINGS IN
HOMOGENEOUS SOILS
3.2 Footing on Homogeneous C-^ Soil 38
3.2.1 The Problem 38
3.2.2 The Objective Function 39
3.2.3 The Boundary Conditions 39
3.3.4 Results and Discussions 39
3.3 Footing on Cohesive Soil 49
3.3.1 The Problem 49
3.3.2 The Objective Function 50
LIST OF FIGURES
Figure Page
2.1 Discretization of the soil mass for a typical problem 23
2.2 Definition sketch and body forces for rith element 24
2.3 Internal stresses at point i 25
2.4 Continuity of nodal stresses 28
3.1 Strip footing on homogeneous C ~ 4> soil 38
3.2(a) Variation of objective function with penalty parameter 44
3.2(b) Variation of objective function with number of function evaluations 44
3.3 Strip footing on cohesive soil 49
4.1 Details of surface strip footing on two layered soil deposit 54
4.2 Mesh patterns for footings on two layered soil deposits 55
4.3(a) Variation of objective function with penalty parameter 61
4.3(b) Variation of objective function with number of function evaluations 61
4.4 Studies on extensibility for case 3 63
LIST OF TABLES
Table Page
3.1 Optimization details for the bearing capacity problem 40
3.2 Final design vector, sigma vector, constraints and objective function 42
value for eighteen elements
3.3 Stress field and stress-strength ratios at the nodal points 45
for eighteen elements
3.4 Comparison of bearing capacity solutions 47
4.1 Comparison of bearing capacity factors for footings on 47
.two layered soil deposits
4.2 Final design vector, sigma vector, constraints and objective function 59
value for case 3
4.3 Stress field and stress-strength ratios at the nodal points 62
for case 3
NOTATIONS
flj = Coefficient to <Tj in linear function to be optimized.
а, j = Coefficient to in linear constraint number i.
[j 4] = Coefficient matrix of the linear equality constraints.
б, , 6 = Coefficients.
B = Width of footing.
[jB] = 9x7 matrix, geometrical property of the element.
Cu = Undrained cohesive strength of the soil.
C = Cohesion of the soil.
{D} = Design vector.
Dm = Optimum design vector.
6 = Angle of surface friction.
St = Transition term between two types of penalty terms.
F(D) = Objective function.
[G] = 9x7 matrix, geometrical property of the element.
{g} = 9 component vector related to body forces inrith. element.
= Inequality constraints.
H = Depth of the top layer of soil deposit.
{h} = 9 component vector related to body forces inuth element.
M = Total number of inequality constraints.
m,n = Element numbers,
rjt = Penalty parameter.
[5] = 7x9 matrix, geometrical property of nth element.
{s,} =3 component internal stress vector at node i of element n.
s =9 component stress vector which defines internal stresses in 7^^A element.
[T] = 6x9 matrix, geometrical property of nth element.
X,
= X coordinate of nodal point i.
Zj = z coordinate of nodal point j.
Ixilz = Body forces per unit volume in x and z directions.
= Slope of element side connecting nodal points i and j.
/i, e = Known coefficients.
<^x,i = Normal stress on vertical plane through nodal point i.
— Normal stress on horizontal plane through nodal point i.
= Normal stress at nodal point i on plane parallel to element side jk of element n.
cr," = Normal stress at point i of element n on side ij.
<7 = Stress vector which defines the complete stress field.
cr" =7 component stress vector which defines external normal stress on rith element.
— Shear stress on horizontal plane at nodal point i.
T,j = Shear stress at node i on the side connecting points i andj.
r" =6 component stress vector which defines the external shear stresses on tlth element.
<j) = Angle of internal friction.
CHAPTER 1
INTRODUCTION
1.1 General
In recent years lot of interest has been generated among the researchers in the area of
geotechnical engineering to find the lower bound limit load for stability problems. The
theoretical foundation of limit analysis hcis been under lain by Drucker, Greenberg and
Prager (1952). For the materials with an associated fiow rules, useful limit theorems
(upper and lower bounds) can be applied to approximate the critical load, even if it
cannot be determined exactly. An upper bound is often a good estimate of the collapse
load but the lower bound is more important as it results in a safe design. However,
except in few cases, it has not been possible to construct the statically admissible stress
field for gravity loaded soil problems. To asses the state of the art a brief review
pertaining to the work was carried out and is presented as follows.
1.2 Brief Review of Literature:
Since Coulomb first published his classical earth pressure theory in 1776, lot of
development has taken place in soil plasticity over the years. However, only after 1952
when Drucker and Prager extended the study of Drucker et al. (1952) for perfectly
plastic materials which obey Mohr-Coulomb yield criterion to granular material, limit
analysis has extensively been used. Apart from limit analysis other methods that are
commonly used to estimate the critical load of a foundation are limit equilibrium.
method of characteristics and finite elements.
2
The review pertains mainly to the methods employed in determining the bearing
capacity of footings resting on homogeneous and stratified soil deposits and leaves out
many other details. For validation of the predictive models it is necessary to have
experimental data and, as such, a few such work pertaining to bearing capacity have
also been reviewed.
Button (1953), for the first time, analyzed the bearing capacity of continuous footing
on two layered soil deposits using limiting equilibrium method. He has assumed general
shear failure along the cylindrical slip surfaces starting at the edge of foundation and
presented modified bearing capacity factors A^cm for saturated clays under undrained
(<j)^ = 0) conditions with various values of C 2 IC 1 ; <l>u is the angle of shearing resistance
under undrained conditions and Ci and C 2 are the undrained shear strengths of the
top and bottom layer respectively. Two cases have been considered : (a) the shear
strength, in each layer is constant with depth and (b) the shear strength of the upper
layer decreases or increases with depth to a value C[ and the lower layer has a constant
strength C[ with depth. Results have been presented in the form of design charts.
Tcheng (1957) has conducted tests for determining the bearing capacity of shallow
foundations on a stratified soil deposit. The supporting soil consists of two layers
with sand in upper layer of finite thickness and soft clay at the bottom being infinite.
Experimental studies show that the mode of failure is punching along essentially vertical
slip lines emanating from the foundation perimeter when the thickness of the top layer
is less than 1.5 times the width of the footing. Based on model tests and theoretical
3
analysis of rupture surface observed, Tcheng has proposed empirical formulae for long
rectangular footing resting on two layer soil system described above and has shown that
the influence of the soft clay layer on bearing capacity becomes negligible when the sand
layer thickness exceeds 3.5 times the footing width. The results obtained compare very
well with the test results on the model.
Sokolovsky (1960, 1965) extensively used the method of characteristics to predict
the lower bound bearing capacity and earth pressure problems; he also used the same
method to study the stability of slopes. He considered homogeneous soils obeying
Mohr-Coulomb failure criterion.
Yamaguchi (1963) has investigated the bearing capacity of a sandy layer of finite
thickness resting on a soft clayey layer assuming a dispersion angle for pressure in sand
layer below footing and taking uniform pressure at the top of the clay layer. He has
presented expressions for bearing capacity values and has also discussed the method
or principle to improve the ground economically. It has been shown that for small
footings the top layer governs the bearing capacity values whereas for footings of large
diameters clay layer is the controlling factor. He concludes that strip foundation is more
economical when the sandy layer is firmer than clayey layer whereas raft foundation is
preferable when the strength conditions are reversed. He also concludes that sand drain
is better for improving two layered ground especially when sandy layer is in loose state.
Finn (1967) has presented a limiting plasticity theory on the basis of Mohr-Coulomb
yield criterion and associated flow rule to provide upper and lower bound solutions to
4
problems in soil mechanics. It has been assumed that the soil reaches a perfectly plastic
state and that no volume contraction occurs during plastic deformation. To illustrate
the principles of the theory, it has been applied to classical problems of soil mechanics
viz. critical height of vertical cut, pressures on retaining walls, ultimate bearing capacity
of footings and bearing capacity of footings on slopes.
Using limiting equihbrium method, Siva Reddy and Srinivasan (1967) have further
extended the work of Button (1953) to consider the non-homogeneity and anisotropy of
soil with respect to shear strength. They have studied the effect of degree of anisotropy
on bearing capacity for both the cases considered by Button (1953). For values of
degree of anisotropy if > 1, the ultimate bearing capacity is smaller than that for
isotropic medium with constant vertical shear strength whereas for values K < 1 the
ultimate bearing capacity is greater for anisotropic soils. The numerical results have
been presented in the form of graphs for various degrees of anisotropy.
Davis (1968) has obtained lower and upper bound solutions under plane strain condi-
tions for material with associated flow rule for ultimate bearing capacity of strip footing
on pure cohesive soils, passive failure of cohesive-frictional soils and pressure on tunnel
roofs overlain by clay by using discontinuous stress and velocity fields. By taking the
problem of unconfined compression between rough end plates, it has been shown that
the use of limit theorem is not justified for materials with non-associated flow rules.
Graham (1968) has investigated a numerical procedure based on the work of Sokolovsky
(1960) to study the failure of retaining walls, slopes and deep strip footings and extended
5
it to take into account of non-homogeneity in the cross-section. The material has been
assumed to be rigid plastic obeying Mohr-Coulomb failure criterion. The results ob-
tained compare very well with the existing theories and test results.
Yokow et al. (1968) have extended Meyerhof’s method (1951) to obtain the ultimate
bearing capacity of a strip footing in two layered ground when the base of the footing is
set in the supporting soil overlain by the weaker layer. The effect of shearing strength of
the weak layer has been included in the bearing capacity analysis. The analysis is based
on the assumption that soil is weightless rigid plastic body obeying Mohr-Coulomb yield
criterion. Method of characteristics has been used to obtain the solutions. Two example
problems have been undertaken to show the applicability of the proposed analysis.
Brown and Meyerhof (1969) have conducted experiments on the bearing capacity
of layered clays using circular and strip footings for a range of layer thickness and
clay strengths. Total stress analysis has been done. For stiff clays overlying soft clays
failure occurs by punching of the footing through the top layer with full development
of the bearing capacity of the lower layer. For the reverse case failure occurs mainly by
squeezing of the top soft layer between footing and stiffer layer below, with more and
more interaction between the layers as the strength ratio approached unity. They have
presented their experimental results in the form of graphs for strip as well as circular
footings resting on layered clays.
Mandel and Salencon (1969) have analyzed the bearing capacity of strip footing on
two layered soil system using method of characteristics. The solution indicates that the
6
presence of a rigid layer below the bearing stratum results in an increase of bearing
capacity.
Belytschko and Hodge (1970), using finite element technique, have presented an
interesting general approach for finding the lower bound limit load for plane stress
problems. Lower bounds have been obtained for a number of weakened slabs and com-
pared with upper bounds obtained by previously available methods. A good agreement
is noticed. The method is also of interest to geotechnical engineers due to its potential
to be extended in solving stability problems.
Chen and Scawthorn (1970) have presented a critical discussion on the significance
of the limit equilibrium and limit analysis solutions. They have shown that within
the framework of idealizations the limit analysis approach is rigorous, competitive with
limit equilibrium and in some instances much simpler. They have analyzed the bearing
capacity of strip footings and the earth pressure problem using classical Coulomb plane
failure mechanism and simple discontinuous stress field. Based on the results obtained.
They have concluded that the assumption of perfect plasticity is very good for stability
problems in soil mechanics.
Desai and Reese (1970) have used finite element method to investigate the behaviour
of circular footings on a single as well as two layers of clay. The method employs
non-linear stress-strain relationship, obtained from triaxial tests, to predict the load
displacement relation of a steel footing. The results obtained for two layer soil system
are found to be in good agreement with the test results.
7
Lysmer(1970), for the first time, developed a generalized method for lower bound
analysis of plane problems in soil mechanics. The method uses simple three nodded
triangular elements in which stress distribution has been assumed to be linear. Mohr-
Coulomb yield criterion has been used. The problem has been formulated as a linear
programming problem by linearizing the non-linear yield criterion. This has been ap-
plied to several earth pressure and bearing capacity problems. The results obtained
compare very well with known solutions.
Krishnamurthy (1972) extended the method of characteristics to determine bearing
capacity for layered C-(f> soils obeying Mohr-Coulomb criterion of general shear failure.
The values of cohesion, angle of internal friction and unit weight in each layer have been
used to obtain stresses and slip lines. He has used finite difference technique to solve
the differential equations in a manner similar to that of Sokolovsky’s approach. Three
different combinations of C and <^(o)C 2 /Ci = 2.0, = 0.75(6)C2/Ci = 4.0, —
0.6 and (c) C^jCi = 0.4,<^2/<?^i = 1-25 have been analyzed. He has also obtained the
solutions for inchned loads on both homogeneous and layered soils. Results have been
presented in the form of design charts.
Mandel and Salencon (1972) have obtained solutions for the bearing capacity of
a soft ground layer overlying a rigid base using the theory of limiting equilibrium for
plane strain conditions. Results have been obtained for the material obeying Coulomb’s
yield criterion for 0° < (^ < 40°. Effect of base friction and the ratio 5/h(J5=width of
footing, h=depth of the top layer) have also been considered in the analysis and design
8
charts have been presented. It has been shown that for a perfectly rough contact the
bearing capacity, starting from the classical value, increases steadily with Bjh whereas
for perfectly smooth contact the same decreases from the classical value, reaches a
minimum and then in dealing with wide foundation it increases, becoming greater than
the classical value.
Sabzevari and Ghahramani (1972) have presented an analytical study concerning
the limit equilibrium of non-homogeneous soil medium satisfying non-linear yield cri-
terion. Method of characteristic has been used in the analysis to derive the recurrence
formillae. This has been applied to bearing capacity and earth pressure problems. The
results obtained have been compared with those predicted by conventional theories of
homogeneous soils. A significant difference between these two results show that the
slip line fields as well as the stress distributions for bearing capacity and earth pressure
problems in non-homogeneous soils with non-linear failure criterion, cannot be deter-
mined accurately from conventional limit equilibrium approach even if the analysis is
based on the average values of cohesion, angle of internal friction and unit weight.
Chen and Davidson (1973) have obtained the upper bound limit load for both surface
and embedded footings with smooth and rough bases. The soil is modeled as an elastic
perfectly plastic material obeying Coulomb yield criterion. The analysis presented
indicates that the significance of base friction is greatly reduced for deep footings.
The results obtained compare well with existing solutions for both smooth and rough
footings.
9
Davis and Booker (1973) have obtained upper bound solutions to problems of bear-
ing capacity of clay which is inhomogeneous in vertical direction only. They have shown
that the rate of increase of cohesion with depth plays the same role as density plays in
the bearing capacity of homogeneous cohesive frictional soils. They have shown that for
rigid footings, the bearing capacity depends upon the breadth and also that the rough-
ness of footing may have small but significant effect in increasing the bearing capacity
in contrast to the homogeneous case for which roughness has no effect. Results have
been compared with those of slip circle analysis and it has been shown that the slip
circle' solutions may very seriously overestimate the bearing capacity of rigid footings.
Mayerhof (1974) has investigated the ultimate bearing capacity of both circular and
strip footings resting on subsoils consisting of two layers for the case of dense sand
on stiff clay and loose sand on stiff clay. The obtained results for different modes of
soil failure have been compared with the results of model tests on circular and strip
footings and some field observations of foundation failures. They have shown that the
ultimate bearing capacity of footings on sand layer overlying clay can be expressed by
punching shear coefficients for the case of dense sand on stiff clay and by modified
bearing capacity coefficients for the case of loose sand on stiff clay. Theory and test
results show that the influence of the sand layer thickness beneath the footing depends
mainly on the bearing capacity ratio of the clay to sand, the friction angle of sand, the
shape and the depth of the foundation.
Purushottamaraj et al. (1974) have presented upper bound limit analysis approach
10
for determining the ultimate bearing capacity of footings on two layered soils. They have
considered the failure mechanism fundamentally similar to that of Prandtl-Terzaghi
mechanism but with a different wedge angle. The critical wedge angles have been found
in each case. However, they have presented bearing capacity charts for footings by
varying only cohesion in layers and keeping the friction angle and unit weight constant.
Basudhar (1976) and Basudhar et al. (1979,1981) modified Lysmer’s approach
(1970) by incorporating the non-linear no-yield condition constraints directly in the
analysis, thus formulating the lower bound optimization problem as non-linear pro-
gramming problem. The constrained optimization problem has been converted to an
unconstrained one using the extended penalty function method as suggested by Kavlie
and Moe (1971). The sequential unconstrained minimization of the composite function
so developed was carried out by using Powell’s method along with quadratic interpola-
tion technique for multidimensional and unidirectional search respectively (Fox, 1971;
Rao, 1984). The method has been applied to bearing capacity and earth pressure
problems. Results obtained compare very well with those of Lysmer’s (1970).
Gioda and Donato (1979) have presented a numerical procedure based on finite
elements and mathematical programming technique for the solution of geotechnical
problems where elastic-plastic material behaviour is considered. The proposed approach
can be adopted for geotechnical media characterized by any suitable yield condition,
accounting, if necessary, for work hardening behaviour. Three geotechnical problems
viz. determination of surface settlement produced by a strip load acting on a layered
11
soil deposit of finite thickness, horizontal and vertical displacement caused by an open
excavation in a layered soil deposit and the surface settlements, linear deformation
and stress states after the completion of shallow tunnel excavation have been dealt
with to show the applicability of the proposed procedure. The results obtained have
been compared with in-situ measurements and other available results. A reasonable
agreement has been noticed.
Bottero et al. (1980) have presented an elasto-plastic finite element formulation
using limit analysis theory to obtain lower and upper bounds of plane strain problems
in soil mechanics. The problem has been formulated as a linear programming problem
by using a linearized yield criterion for standard Tresca material with linear variation
in stress and velocity fields. The problems of ultimate bearing capacity of strip footing,
pull out capacity of foundations and slope stability have been dealt with to show the
efficiency of the two proposed procedures.
Satyeinarayaiia and Garg (1980) have proposed an empirical method to predict nu-
merically the ultimate bearing capacity of footings on layered soils. They have given
expressions for average values of shear strength parameters C and 4 for the two layered
system which can be used directly in the classical bearing capacity equations. The
computed values are found to be in reasonable agreement with experimental results.
Hanna (1981) has conducted an experimental investigation to examine the validity
of the method proposed by Satyanarayana and Garg (1980) for bearing capacity of strip
and circular footings on two layered soils. He has concluded that more refinement and
12
further experimental and possibly field verifications are needed before recommending
its implementation for practical purposes.
Kusakabe et al. (1981) have obtained the bearing capacity solutions of slopes loaded
on top surface using upper bound theorem. The results have been compared with those
obtained by conventional circular arc methods as well as by Kotter’s stress characteristic
equations. They have concluded that upper bound is useful from the engineering point
of view because of the simplicity of the method. To check the validity of the upper
bound solutions, model tests have also been conducted. The model tests show that the
theory underestimates the bearing capacity. The failure mechanisms predicted by the
theory with —Q assumption are in reasonable agreement with observation in model
tests. Lysmer’s (1970) method has also been used to obtain lower bound solutions to
asses the validity of the upper bound analysis. The upper bound solutions are shown
to be good approximation of exact solutions for bearing capacity of loaded slopes. The
computed results are presented in the form of charts.
Caciaro and Cascini (1982) have proposed a mixed variational principle for the limit
analysis of perfectly plastic continua in which the non-linear yield criterion and the
associated flow rule appear through a ’penalty’ function. Using mixed finite element
discrete formulation^ and sequential unconstrained minimization technique, they have
presented several numerical results for both structural mechanics and soil mechanics
problems and have compared them with previously available exact and numerical solu-
tions. A close agreement is noticed.
13
Hanna (1982) has investigated the ultimate bearing capacity of footings resting on
subsoils consisting of a weak sand layer overlying a strong deposit. Based on model tests
of strip and circular footings, he has shown that the bearing capacity of a weak sand
layer overlying a strong deposit can be expressed by the classical equation of bearing
capacity for homogeneous sand in conjunction with modified bearing capacity factors.
The theory compares well with the available model test results. Design charts have
been presented.
Baker and Frydman (1983) have studied the problem of finding the bearing capacity
of a strip footing resting on the upper surface of a slope and have discussed the effect
of non-linearity in the failure criterion of soil on the upper bound solution procedure.
By considering the inherent non-linearity of the failure criterion, it has been shown
that the upper bound solution procedure yields not only the minimum value of and
the external load and the failure mechanism but also the stress distribution along the
slip surface. They have demonstrated that there is a fundamental difference in the
procedure used for applying the theorem to materials with linear and non-linear failure
envelopes, which they have concluded to be due to the different roles played by the
normality criterion in these two cases.
Baus and Wang (1983) have investigated, experimentally and analytically, the bear-
ing capacity of footings located above a continuous void in silty clay soil. The analysis
has been done by finite element method treating the soil as an elastic perfectly plastic
material. Within the elastic range, the stress-strain relationship of the soil is described
14
by Hooke’s law beyond which, the soil is as perfectly plastic in accordance with Von
Mises yield criterion. It has been demonstrated that, for practical purposes, the void
shape has negligible effect on the bearing capacity. Results also indicate an increase in
bearing capacity with increasing depth of foundation when the depth of void is main-
tained constant. All the results have been presented in the graphical form.
Mizuno and Chen (1983) using finite element formulation and adopting Drucker-
Prager models with associated as well as non-associated flow' rules and cap models have
obtained solutions for problems of flexible smooth and rough rigid footing resting on an
over consolidated stratum of clay. They have observed that the velocity fields predicted
by the plane cap model for both type of footing problems do not agree with that of the
Prandtl’s solution in the ’radial shearing zone’ and ’near the surface zone’, but, that
predicted by Drucker-Prager and elliptic cap model agree well with Prandtl’s solution
for both the footing problems.
Reddy and Rao (1983) have obtained the upper bound bearing capacity of a strip
footing on a two layer C-4> soil system exhibiting anisotropy and non-homogeneity in
cohesion assuming Prandtl-Terzaghi failure mechanism with varying boundary wedge
angles and presented the results in the form of non-dimensional charts. It is noticed that
anisotropy and non-homogeneity in cohesion in each layer have considerable influence
on the ultimate bearing capacity.
De Borst and Vermeer (1984) have examined the ability of a 15 nodded displacement
type finite element to obtain the critical loads of soil structures for soils with high
15
frictional angle and with non-associated flow rules. Solutions have been presented for
strip and circular footings, for the trap door problem and for the cone penetration test.
With reference to footing problems, the accuracy of the numerical solution has been
shown to be very high but stability problems occur when non-associated flow rules are
applied.
Tamura et al. (1984) have investigated a numerical procedure to ancdyze the limit
state of soil structures assuming the soil to be rigid plastic. The rigid plastic finite ele-
ment method has been formulated on the basis of upper bound theorem. The numerical
procedure has been investigated by typical problems viz. bearing capacity of shallow
foundation and slope stability. Good agreement between the results and the existing
solutions has been noticed.
Aral and Tagyo (1985) have developed a numerical procedure that furnishes a rea-
sonable lower bound solution for the problems of bearing capacity and slope stability
analysis. The stress field is discretized into quadrilateral elements and the formulated
optimization problem is solved numerically using non-linear programming and sequen-
tial unconstrained minimization technique. It has been proved that the procedure
provides an appropriate and stable lower bound solution for general soils which have
cohesion, friction angle and its own weight, so far as the friction angle is not so large.
However, the procedure cannot represent the arbitrary stress conditions at the boundary
surface because a set of stresses is assumed to be constant within each element. Also,
the procedure is difficult to apply for problems of soil structure interaction viz. earth
16
pressure problem since the procedure considers the stress as the independent variable
and assumes the soil ma^s to be rigid perfectly plastic material. The validity of the
procedure is successfully demonstrated through several case studies.
Tamura et al. (1987) have developed a rigid plastic finite element method for fric-
tional materials. The stress-strain rate relation for a rigid plastic material of Drucker-
Prager type under the assumption of associated flow rule has been derived. They have
observed that materials with high friction angle values show somewhat unreasonable
velocity field due to the dilatancy effect affecting the bearing capacity solutions. As
such, ‘a numerical technique for the non-associated flow rule to reduce such effects has
beeti developed by satisfying both the yield condition and the normality for the plastic
potential.
Sloan (1988) modified the method of Bottero et al. (1980) to obtain the lower bound
solution for strip footing under plane strain conditions. A perfectly plastic soil model
has been assumed, which may be either purely cohesive or cohesive frictional together
with an associated flow rule. Mohr-Coulomb yield criterion has been assumed, the liner
approximation of which enables the formulation to compute statically admissible stress
field via finite elements and linear progra m min g . Active set algorithm has been used
to solve lower bound optimization problem which makes the method appreciably faster
than than the displacement type of finite element method for predicting collapse load.
He has solved bearing capacity problems of strip footing for homogeneous soil as well
as for a purely cohesive soil which has increasing strength with depth. The obtained
17
solutions compare very well with the available Prandtl’s and exact solutions.
Reddy et al. (1989), using the method of characteristics, obtained the bearing
capacity factors for a circular footing placed at the interface of a two layered soil with
the top layer being weaker than the bottom layer. The ground surface is taken to be
horizontal up to a certain distance from footing beyond which it has been assumed to
be inchned. The numerical results presented show that the bearing capacity factors are
influenced by the stratification, strength of the soil layers and the depth at which the
footing is placed and to a lesser extent by the other parameters.
Sloan (1989), assuming a perfectly plastic soil model which is either purely cohesive
or cohesive frictional, has adopted the finite element formulation in conjunction with
the upper bound limit theorem. It has been shown that the upper bound optimization
problem may be solved efficiently by applying an active set algorithm to the dual linear
programming problem. Upper bound solutions for strip footing as well as for a trapdoor
in a purely cohesive soil have been obtained. These solutions compare very well with
the available solutions for corresponding problems.
Reddy et al. (1990) used the method of characteristics to estimate the bearing
capacity of strip footing placed at the interface of two layered soil with the bottom
layer stronger than the top layer when the ground has an upward linear slope at a
distance from the footing. The results presented show that the presence of an upward
slope just adjacent to the footing and the presence of a stronger layer below the base
of the footing increases the bearing capacity considerably.
18
Azam et. al. (1991) studied the performance of a strip footing on homogeneous and
stratified soil deposits containing two soil layers both with and without a continuous
void. They have used two dimensional finite element method and predicted the collapse
load. To accommodate the non-linear stress strain characteristics of the foundation soil
in the finite element analysis the incremental footing load is applied. For stratified soils,
the obtained results compare well with the solutions of Vesic (1975).
Yong and Mohamed (1991) have developed an analytical method using FEM and
non-lineax stress analysis for predicting the performance of a muskeg deposit under
loading. The deposit has been modeled as layered system consisting of three layers
(surface mat, peat layer and mineral soil). The analytical results are found to be in
good agreement with experimental results.
Chuang (1992) formulated the limit analysis of stability problems in geomechanics
as a pair of primal-dual linear programs. The formulation provides a solution that
is claimed to be both kinematically and statically admissible. For an assumed finite
element mesh, the solution identifies the critical collapse mechanism among all the pos-
sible failure mechanisms contained within the given mesh and gives the corresponding
values of both static and kinematic variables, together with the critical load parameter.
Numerical solutions to bearing capacity problems as well as slope stability problems
have been obtained. The method can readily handle failure surfaces of any arbitrary
shape, external forces acting on the soil mass with varying pore water pressure, tension
cracks filled with water and inhomogeneous material having both cohesion and angle
19
of internal friction. The computed results compare very well with the corresponding
values of the analytical and numerical solutions.
Singh (1992) studied a number of stability problems in geotechnical engineering
using Lysmer-Basudhar approach (Lysmer, 1970; Basudhar, 1976). For some of the
problems he could compare his results with the lower bound solutions using method
of characteristics, finite element and linear programming etc. The results obtained
have been found to be in good agreement with the available results for most of the
corresponding problems.
1.3 Motivation of the Work:
It can be seen from the reviewed literature that in nineteen hundred sixties and
even thereafter, method of characteristics as suggested by Sokolovsky (1960, 1965) has
been predominantly used by the research workers to predict the lower bound limit loads
of stability problems in geotechnical engineering. But, in the early phases, solutions
were available only for homogeneous and isotropic materials. In the early seventies
efforts were made to extend these work to non-homogeneous and stratified deposits
(Krishnamurthy, 1972; Purushottamaraj et al. 1974). The method of characteristics
generally becomes very complicated for complex problems. As such, the need for a
more generalized method to construct statically admissible stress field was felt by the
research community. Thus a varity of methods combining the flexibility of finite ele-
ment methods and the elegance of optimization technique in isolating the optimal lower
bound limit load have been developed. In this direction the pioneering work of Lysmer
20
(1970), Bottero et al. (1980), Munro (1982) are worth mentioning. Another method
which needs special attention is that of Hodge (1970); even though this method was
developed for analyzing plate problems, the method is of interest to geotechnical en-
gineers for the general nature of the solution procedure and its potentiality to solve
stability problems. Apart from the developments of new methods to construct the stat-
ically admissible stress field, efforts have been continued over the last two decades to
apply more and more sophisticated algorithms to enhance the computational efficiency
of the original methods. With this in view Basudhar (1976) modified Lysmer’s method
and formulating the problem as a non-linear programming one, isolated the optimal
stress field. However, the methods did not find much appreciation as these were con-
strained by the non-availability of high speed digital computers. In the eighties, with
the revolutionary break through in the computers, new interest had been generated in
applying these techniques. It is evident by the fact that after 1980 it was in 1985 that
Arai and Tagyo made an effort to introduce quadrilateral elements and used non-linear
programming technique to isolate the optimal stress field. In 1988 Sloan made an effort
to improve upon Bottero et al’s. approach (1980). Sloan and Asadi (1991) then used
the same technique to a new class of problem namely the trapdoor problem. Side by
side since 1989, research has been pursued at 1. 1. T. Kanpur, to apply Lysmer-Basudhar
approach (Lysmer, 1970; Basudhar, 1976; Basudhar et al. 1979 and 1981) to different
class of stability problems and the outcome has been reported by Singh (1992), Singh
and Basudhar (1992,1993a and 19936).
21
So it is evident that various new methods of analysis are increasingly being suggested
to predict the lower bound limit load for stability problems. As such, there is a need
to asses the strength and weaknesses of these methods and make a comparative study.
In addition there is also a need to extend these methods to new areas and, if possible,
calibrate the models by comparing the obtained results with experimental values and
also with other solutions available in the literature. With this in view an effort has
been made in this thesis to make a comparative study of Lj'^smer-Basudhar approach,
Arai and Tagyo approach, Bottero-Sloan’s approach and Munro-Chuang approach. In
addition Lysmer-Basudhar approach has further been extended to find the bearing
capacity of strip footing resting on the surface of two layered soil deposits.
The comparison of the predicted results using Lysmer-Basudhar approach with both
experimental and theoretical values reported in literature enables one to judge the
capability of the method used in the present study vis a vis other methods.
1.4 Scope and Organization:
In chapter 2 the original formulation of the Lysmer-Basudhar approach adopted in
this thesis for analyzing the stability problems has been presented in brief.
In chapter 3 a comparative study of the different techniques has been undertaken
and presented with reference to a smooth strip surface surface footing resting on homo-
geneous, general C-4> soil.
In chapter 4 Lysmer-Basudhar approach has been extended to find out the bearing
capacity of both rough and smooth surface strip footings resting on a two layered soil
JH
22
deposit. A study regarding the extensibility of the stress field has been presented.
The obtained results have been compared with available experimental and numerical
solutions reported in the literature.
Generalized conclusions and scope of future work has been presented in chapter 5.
In the appendix, a mathematical proof, as given by Lysmer (1993), of the fact that
satisfying the no-yield condition only at the nodes of the triangular elements is sufficient
to ensure that yielding does not occur at any point in the element, has been presented.
CHAPTER 2
GENERAL METHOD OF ANALYSIS
2.1 General:
The generalized method of lower bound limit analysis as developed by Lysmer (1970)
and subsequently modified by Ba^udhar (1976) to incorporate the non-linear no-yield
condition constraints directly in the analysis has been adopted for analyzing the prob-
lems.
The method generates stress fields which are in equilibrium everywhere and do
not violate the Mohr-Coulomb failure criterion at any point inside the soil medium.
Furthermore since infinitely many stress fields satisfy these conditions for any given
problem, the method is formulated as a mathematical programming problem to isolate
stress fields which yield high lower bounds. The stress field that is considered in this
method has the property that all stresses vary linearly within each element of some
mesh which cover the soil mass under study. For the sake of completeness the method
is presented herein in brief.
Fig 2.1 Discretization of the soil mass for a typical problem
24
The first step in the analysis of a typical problem, such as the bearing capacity
problem shown in Fig 2.1, is the disretization of the soil mass under consideration
into a mesh of finite number of triangular elements. If possible, the zone of influence
considered for discretization should be based on previous experimental and theoretical
studies. Further discretization of this zone should be done keeping in mind the guide
lines suggested by Lysmer (1970). All nodal points, elements and element sides are
then numbered in some arbitrary order. It can be shown that a mesh consisting of p
elements connecting at q nodal points will have p + q — \ element sides.
2.2 Element Equilibrium:
The geometry of a typical element n, and the external stresses and the body forces
acting on this element are shown in Fig 2.2.
Fig 2.2 Definition sketch and body forces for n^h element
The stresses are assumed to vary linearly within each element, hence the stresses
only at the nodes are considered. In addition, one internal stress cr" is defined as the
normal stress at node i acting on a plane parallel to the side jk. The normal stresses
shown in Fig 2.2 are collected into a 7-component stress vector {a"} defined by
25
IWVV = {aV,fcO-y<Tj.ajfcajtjOri.}
( 1 )
and the external shear stresses are collected into a 6- component stress vector {t"}
defined by
= {'^tkr,jr^irjkrk,rk,} ( 2 )
the internal stresses in each element are collected into 9-component stress vector {s}
defined as
{5} —
(3)
with, {5i}^ = etc.
(4)
Fig. 2.3 Internal stresses at point i.
where, { 3 ^} are the internal stresses at node i. The equilibrium conditions for the
infinitesimal triangle at node i shown in Fig.2.3, are expressed in terms of Uit and
26
as:
Cxk — sin cos Ofk '^zx,x sin 2^j/c (6)
rik = 0.5(crx,, — (7i,,)sin20,fc -f- tzx,zcos20,A: (6)
Similar equations are written for nodes j and k and substituted in eqns. (1) and (2)
to yield
= 1«1
(7)
{r)" = (T)
(8)
Matrices [5] and [T] are the geometric properties of nth element. The conditions of
internal equilibrium are
da^
^'^ZX
dz
J
dx
= Iz
dr^x
, dox
dz
+ 97
— 'lx
(9)
To satisfy these relations, the linear stress fields within the nth element are expressed
in the following form:
<7j = Ci2 + C2a; + C3 + 7^2
CTx = C4Z + C5X + Ce -f 7^3:
Tzx =
-C5Z - Cio: + C7
( 10 )
27
Thus the stress field depends on seven parameters c, which may be combined into
the vector
{cY = {ci c^ cz C 4 C 5 ce ct } (11)
Using eqns. (3), (4) and (10) the stress vector {s} can be written as:
{»} = !(?l{c) + Is) (12)
where
{aV = {izZt, 7xa:,, 0, 7 * 2 ^, 7 ra-j, 0, Ix^k, 0} (13)
is purely a function of geometric properties of nth element. Using equns. (7) and
( 12 ),
{c} = ([5][G])-'{o-}" - ([5][G])-M5]{5} (14)
which when substituted into eqn. ( 12 ) gives
w = IFIW" + {A} (15)
where \B] = (G]((S1[G))“‘ and {h] = (j) - (B][5){s) (16)
Both [jB] and {h} are geometric properties of nth element. Using eqns. ( 8 ) and (15),
we get
28
= miBH.’)” + mu) (17)
which is the equilibrium condition for nth element.
2.3 Interface Equilibrium:
The elements of {cr}" vectors for all the elements are collected into a general {<t}
vector. A system consisting of p elements connected at q nodal points will have (3p -f
2g — 2) stress variables in {a} vector, which are the principal unknowns.
The continuity of normal and shear stresses across any interface as shown in Fig.
2.4 requires:
I
Fig 2.4 Continuity of nodal stresses
for all corresponding values of i,j,m and n. These conditions yield a set of linear
equality constraints in terms of the principal unknowns. The total number of interface
equilibrium condition is equal to twice the number of element sides in contact.
29
2.4 External Boundary Conditions:
The boundary stresses on the external faces of the system may be expressed either as :
T.j < na,j
cr,j = ( and I or r,j = ecr.j (19)
Equns. (18), and (19) can be transformed into the form :
(3P+29-2)
^ a.jCTj = (20)
j=i
and/or
(3p+29-2)
^ ( 21 )
j=i
2.5 No-yield condition:
For static admissibiHty, the stress field should not violate the Mohr- Coulomb yield
criterion at any point in the soil medium. Since all stresses are assumed to vary linearly
within each element, it is sufficient to satisfy the no yield condition at the element
corners only. The condition at the node i can be written as:
+ (2x^3,,,)^ < [(cTj^,- -f ar,t)Sm<l) -|- 2C Cos<?i']^ (22)
Eqn. (22) is expressed in terms of the principal unknowns as follows:
0 - 2 , . = Z,{s}, = ^,{ 5 } and = T,{s}
(23)
30
where,
Z,=
X.=
T,=
similarly for nodes j and k ,
T,=
and
Tk =
Equns. (21) and (22) yield;
( 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 )
( 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 )
( 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 )
( 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 )
( 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 )
( 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 )
( 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 )
( 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 )
( 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 )
(24)
(25)
(26)
(A{s})^ -f {2Ti{s}f - (jB.{s} Sin^ + 2C Cos(pf < 0 (27)
where,
(Z.-A'-i)
31
B,= {Z, + X-i) (28)
Now, equn. (27) can be rewritten in terms of as:
[Ai ([5]a" + hf + [2ri ([5]cr" + /i)]2
-[B,{[B]a^ + h)sm(l> + 2Ccos<f)f <0 (29)
Similar relations can be obtained for the nodes j and k. The elements of {< 7 }"
vector can be picked up from the general stress vector {cr}. The total number of non-
linear equality constraints will be 3p. It is sufficient to satisfy the non-linear equality
constraints only at the nodal points of the triangular elements to ensure that there is
no yield at any point within the element. The proof of the same as provided by Lysmer
(1993) is given in the appendix.
2.6 Objective function, Design Variables, Design Restrictions and Reduction
of Design Variables:
Since in general infinitely many stress fields will satisfy the aforementioned condi-
tion of static admissibility, the isolation of the stress field which optimizes the objective
function is important. In almost all the problems, the stress quantity is a linear combi-
nation of surface stresses £r,j and Using eqn. (17) this quantity can be transformed
into a linear combination of principal unknown which are termed as design variables.
The problem can be stated as:
OPTIMIZE
]
(30)
32
The design restrictions are interface equilibrium and the external boundary conditions.
As soil cannot take tension, the following constraints are also introduced,
— cTj < 0 (31)
eqns. (20) and (28) are presented in general term as:
g, < 0 (32)
The equality constraints (Eqn. 20) can be rewritten in matrix notation as
[A]{o} = {6} (33)
Some of the elements of {c} vector are specified at the boundary. The following relation
|A-Hcr*} = {(-•} (34)
can be arrived at by eliminating the columns of [A] matrix corresponding to the known
elements of the {<t} vector, {ct*} is a vector which is achieved by eliminating the known
elements of {cr} vector. {6*} vector is calculated as follows:
{b*} = {6} - [A']{a'} (35)
[A'] matrix contains the columns that are removed from [A] matrix and {cr'} contains
those elements of {cr} vector that are specified.
33
The following steps are performed for the general rectangular matrix [A*]:
Stepl. The rank and the linearly dependent rows and columns if there be any of the
given matrix, are determined.
Step2. A sub matrix of maximal rank is expressed as product of triangular factors.
Step3. The non basic rows are expressed in terms of the basic ones.
Step4. The basic variables are expressed in terms of the free variables.
By considering these free variables as design variables and expressing the remaining
basic variables in terms of these design variables the equality constraints (Eqn. 33) are
implicitly satisfied. Such a technique helps in reducing the complexity of the problem
by eliminating the equality constraints and there by reducing the dimensionality of the
problem. The independent design variables so obtained are collected in D vector.
The rank(r) is determined using the standard Gaussian elimin- ation technique with
complete pivoting. This implies that the rows and columns of the given m'xn' matrix
[A1 are interchanged at each elimination step if necessary. In general the following
cases may arise:
1. r = m' — n'
[A*] is non-singular and [A*]{<t*} = {6*} has uniquely determined solution.
2. r <m’
[A*] is not row regular and the solution of Eqn. (33) exists only if the remaining (m’ — r)
equations are linearly dependent.
3. r < Ti’
34
[A*] is not column regular and the system has no trivial solution.
Cases (1) and (2) may occur combined. The solution if it exits, can be uniquely
determined if r = n’, otherwise, it contains (n’ — r) free parameters.
The basic variables (cr**) are expressed in terms of the free design variable (D) as
follows :
Once the steps 1,2 and 3 of the enunciated reduction process are carried out the Eqn.
(34) is reduced to a form.
[A’l'lcr*}" = {b*y
where, the superscript denotes the rth elimination step and
(36)
[A-]' =
I \
(U,UR)
(37)
VLR/
where, i is a unit lower triangular matrix of dimension rx r.
U is a unit upper triangular matrix of dimension rx r.
LR is of dimension [m.' — r)xr] if the matrix [A*] is row regular'
that is (m’ = r),LR is absent in the final factorization.
UR is of dimension rx{n’ — r); if the matrix [A*] is column regular that is (n’ = r),UR
is absent in the final factorization.
Let {< 7 *}’’ and {b*y be partitioned into
35
D
and
Then for a consistent system of equations
a” = U-^L-^6* + HD (38)
where, H = -U'^UR (39)
and h\ = LUa** -h LURD (40)
b\ = LRL-’6J (41)
In the present thesis subroutine MFGR developed by IBM has been used to perform
the calculations enunciated in steps 1 to 4 of the reduction process.
2.7 Mathematical Programming Problem:
Determination of the minimum value of the objective function subject to the in-
equality constraints as described above is formulated as mathematical programming
problem which is stated as follows:
Find Dm such that
■F'(Dm) = ^^.jCTj is minimum (42)
subject to pj(Dm) < 0
There is no loss of generality even though the problem is cast as a minimization
problem as maximum of a function can be achieved by minimizing the negative of the
function.
36
The constrained problem is converted into an unconstrained optimization problem
with the help of extended penalty function technique as suggested by Kavlie and Moe
(1971). The Sequential Unconstrained Minimization of the developed composite func-
tion is carried out using Powell’s conjugate direction algorithm (Powell, 1964) along
with Quadratic interpolation technique for linear minimization to isolate the optimal
solution. These methods are available in any standard text book on Optimization
(Fox, 1971; Rao, 1984). The composite function (j){D,rk) is developed by blending the
objective function and constraints as follows:
M
^(D,rO = F(D) -f- n X:G[5,(D)] (43)
The function G[5j(D)] is chosen as:
G[5.(D)] =
1/5,(D) 5,(D)<0
2A-ft(D)/A2 S,(D)>A
(43)
where A = —rkfSt
and 8t ■= a constant that defines the transition between the two types of penalty
terms.
In this approach infeasible starting points are readily acceptable to the minimization
algorithm, which makes it a powerful technique for solving various engineering problems
even if an initial feasible design vector is difficult to guess.
CHAPTER 3
LOWER BOUND BEARING CAPACITY OF SURFACE
STRIP FOOTINGS IN HOMOGENEOUS SOILS
3.1 Introduction
The ultimate bearing capacity of strip footings resting on homogeneous soils has
been widely studied by several investigators. The methods of analysis employed are
based on limit equilibrium, limit analysis and finite element techniques. Limit analysis
solutions provide either a lower or an upper bound to the critical load. Chapter 1 of
the thesis gives a brief account of the available lower and upper bound bearing capacity
solutions. It is seen that for isolating the optimal stress field, two approaches viz. linear
programming approach (Lysmer, 1970; Bottero et al., 1980; Munro, 1982; Sloan, 1988;
Chuang, 1992) and non-linear programming approach (Basudhar, 1976; Basudhar et
aJ, 1979, 1981; Arai and Tagyo, 1985) have been employed. But no comparative study
has been taken up to establish the relative merits and demerits of linear and non-linear
programming approaches of finding the optimal lower bound bearing capacity solutions.
As such, a study has been under taken and presented with reference to a smooth strip
footing resting on the surface of homogeneous soils using Lysmer-Basudhar approach.
The obtained solutions are then compared with the values which had been computed
by using linear programming and reported in the literature.
3.2 Footing on Homogeneous C — <f) soil
3.2.1 The Problem
38
Fig. 3.1 shows a smooth strip footing resting on the surface of a general cohesive-
frictional {C — <p) soil. The objective is to determine the bearing capacity factor
when the shear strength parameters C and <f> are 1.00 kPa and 40*^ respectively.
Fig 3.1 Strip footing on homogeneous C — 4> soil
The exact collapse pressure for a smooth strip footing resting on the surface of a
cohesive-frictional weightless soil may be written as
qf = CNc^qN,
where
Nq = exp(7r tan </>) tan^(7r/4 4- (f>/2),
39
= {^q — 1) cot ^
and q is the overburden pressure. From the above relations Ng can be obtained as
follows:
Na =
9/
+ 1
® Ccot<t>
For C = 1.00 kPa and <j) = 40°, Prandtl’s solution for Ng is 64.20; this value has been
reported to be exact by Sloan (1988). This problem was previously solved by Sloan as
a bench mark problem to demonstrate the effectiveness of his technique. The same is
adopted to make a detailed comparative study of the Lysmer-Basudhar approach with
Bottero-Sloan’s approach.
3.2.2 The Objective Function
The objective function is — (cri 2 + < 721 ). Bearing capacity qj is equal to half of the
absolute value of the objective function.
3.2.3 The Boundary Conditions
The boundary conditions for the meshes shown in Fig. 3.1 are
<T12 = <721 — 0
and ri 2 = T 21 = T 23 = T 32 = T 34 — 743 — 745 — 754 — T56 — Tgs — ti_ 2 o — '^20,1 — 0
3.2.4 Results and Discussions
Results were obtained on CONVEX C-220 computer system for different number of
elements (6, 12, 18, 24, 36 and 48) and a convergence study was made to determine the
optimum number of elements. However, only the mesh with eighteen elements is shown
^0
in Fig. 3.1 and the others are not shown for the sake of space and brevity. The mesh
pattern was chosen keeping in mind that the singular points should be common to as
many triangular elements as possible. In Fig. 3.1, point 1 is such a point and all the
eighteen elements have this as a common point. The obtained bearing capacity factors
Ng, total number of principal unknowns, design variables, equality constraints, inequal-
ity constraints and the total number of function evaluations to achieve the optimal
objective function value are presented in Table 3.1.
Table 3.1 Optimization Details for the Bearing Capacity Problem
No. of elements
6
12
18
24
36
48
Ng
47.84
62.42
62.65
62.56
61.91
No. of Principal unknowns
32
62
92
122
182
242
No. of Design variables
13
31
45
63
91
127
No. of Equality constraints
18
46
58
114
No. of Inequality constraints
48
96
144
192
384
No. of function evaluations
23898
76459
226625
416725
The magnitude of problems that are likely to be faced may be imagined from the
data presented in Table 3.1. As the number of elements is increased from six to forty
eight, almost tenfold increase in the number of design variables occured; for meshes with
six and fortyeight elements the number of design variables are 13 and 127 respectively.
In literature (Fox, 1971) it is generally suggested that for problems with more than fifty
design variables, variable matric method should generally be adopted for better stability
Table 3.2
Final Design Vector, Sigma Vector, Constraints and
Objective Function Value for eighteen elements
(D) Vector
4.25026
5.20700
32.1894
16.2139
5 66415
3.98030
9.44989
6.76165
4.12858
4.28736
4.28846
3.99824
3.96229
4.02469
3.97928
4.71694
4.04707
5.72798
5.01296
6.41219
5.30619
7.59059
6.35774
10.6066
8.85594
8.15849
5.75475
17.3116
9.78765
13.6371
12.4178
20.9039
15.0064
19.9580
14.1997
35.8934
15.3154
30.0137
15.4204
6 52259
12.4783
14.4538
3 95233
. 71.2003
75.2055
Sigma Vector
15.4204
15.3154
14.4538
15.9511
32.1894
16.2139
17.3116
12.4783
10.6065
6.52259
9.44989
6 76165
5.66415
5.20700
4.25026
3.95233
3.98030
4.12858
71.2003
75.2055
63.5065
66.9593
49.4223
51.9055
36.0447
37.9901
26.4212
28.1964
18.0344
19.1751
13.2137
14.0551
10.2523
10.8998
8.22595
8.72824
5.37279
5.68797
3.97497
4 21044
3.03096
3.20653
2.28328
2.41456
1.55075
1.64379
1.08422
1.15228
0.71739
0.76838
0.34483
0.37549
0.10042
0.11206
0.00000
0.00000
4.28736
4.28846
3.99824
3.96229
4.02469
3.97928
4.71694
4.04707
5.72798
5.01296
6 41219
5.30619
7.59059
6.35774
10.6066
8.85594
8.15849
5.75475
11.9002
9.78765
13.6371
12.4178
20.9039
15 0064
19.9580
14.1997
35 8934
30.0137
16.1955
16.3270
15.5256
16.1955
15.4205
15.5256
15.4204
15.4205
Interface Shear Equality
Constraints
3.78042E-06
1 00136E-05
-2.61072E-05
-2.81334E-05
1.03166E-05
-9.53674E-07
2.23298E-06
7.62939E-06
6.54843E-06
1.28746E-05
-7.63589E-06
2.62260E-06
1 97856E-06
-4.17233E-06
1.00761E-05
-6.31809E-06
-6.79109E-06
2.50340E-06
-9.04665E-07
4.05312E-06
-1.13633E-06
-3.75509E-06
-3.8731 9E-07
-1.13249E-06
1.85883E-06
4.52995E-06
-2.01272E-06
-'j I'.'jr >'»
-1.06527E-06
1.19209E-07
5.41122E-07
1.19209E-07
-5.55798E-09
-1.49012E-07
. . . contd. on next page
Boundary Shear Equality Constraints
3.63831E-06
6.67572E-06
-6.67572E-06
-3 57628E-06
5.96046E-06
9.53674E-06
-2 86102E-06
4.76837E-06
1.43051E-06
3.81470E-06
-1 62530E-06
-1.34110E-06
Non-linear No-yield Constraints (Inequality)
-1.61687E+02
-3.29590E-02
-5.02930E-02
-1.42879E+02
-5.02930E-02
-2.14006E+01
-3.03296E 00
-2.13948E+01
-1.67326E+02
-1.79947E+02
-1 67328E+02
-1.58709E+02
-8.78906E-03
-8.57813E 00
-8.90161E 00
-4.13330E-01
-2.44141E-02
-4.56653E 00
-6.65283E-03
-7.49512E-02
-2.72687E 00
-9.17023E 00
-1.48639E 00
-1.16593E+01
-6.77490E-03
-9.98444E-01
-5.34058E-03
-5.57388E-01
-1.68182E-01
-7.74551E-01
-4.631p4E-03
-3.74603E-03
-2.29164E-01
-3.63007E-01
-5.28717E-03
-4.90135E-01
.-2.49481E-03
-2.3'.",i2('l (I'j
-1.37718E-01
-8.44784E-02
-5.43594E-03
-8.22334E-02
-4.77600E-03
-2.71988E-03
-1.63841E-03
-1.28166E-01
-9.37843E-03
-3.49255E-02
-8.16898E-02
-4.59862E-03
-3.23677E-03
-4.76498E-01
-2.52533E-03
-1 70517E-03
No-tension Constraints (Inequality)
-1.54204E+01
-1.53154E+01
-1.44538E+01
-1.59511E+01
-3.21894E+01
-1.62139E+01
-L73116E+01
-1.24783E+01
-1.06065E+01
-6.52259E OO
-9.44989E 00
-6 76165E 00
-5.66415E 00
-5.20700E 00
-4.25026E 00
-3.95233E 00
-3.98030E OO
-4.12858E 00
-7.12003E+01
-7.52055E+01
-6.35065E+01
-6.69593E+01
-4.94223E+01
-5.19055E+01
-3.60447E+01
-3.79901E+01
-2.64212E+01
-2.81964E+01
-1.80344E+01
-1.91751E+01
-1.32137E+01
-1.40551E+01
-1.02523E+01
-1 • ( i
-8.22595E 00
-8.72824E 00
-5.37279E 00
-5.68797E OO
-3.97497E 00
-4.21044E 00
-3.03096E 00
-3.20653E 00
-2.28328E 00
-2.41456E 00
-1.55075E 00
-1.64379E 00
-1.08422E 00
-1.15228E 00
-7.17398E-01
-7.68382E-01
-3.44834E-01
-3.75496E-01
-1.00424E-01
-1.12065E-01
-4.28736E 00
-4.28846E 00
-3.99824E 00
-3.96229E 00
-4.02469E OO
-3.97928E 00
-4.71694E 00
-4.04707E 00
-5.72798E 00
-5.01296E 00
-6.41219E 00
-5.30619E 00
-7.59059E 00
-6.35774E 00
-1.06066E+01
-8.85594E 00
-8.15849E OO
-5.75475E 00
-1.19002E+01
-9.78765E 00
-1.36371E+01
-1.24178E+01
-2 09039E+01
-1.50064E+01
-1.99580E+01
-1.41997E+01
-3.58934E+01
-3 00137E+01
-1.61955E+0]
-1 63270E+01
-1.55256E+01
-1.61955E+01
-1.54205E+01
-1.55256E+01
-1.54204E+01
-1.54205E+01
Optimal function value = 146.40
^3
of the numerical scheme. However, due to non-availability of exclusive gradients Pow-
ell’s conjugate direction method, a non-gradient based technique has still been retained
in the Lysmer-Basudhar scheme and computations have been carried out. This adop-
tion hcis worked very well in finding the solutions as has already been discussed, even
with 127 number of design variables. However, there is a four-hundred fold increase in
the number of function evaluations for fortyeight elements with that of six elements. It
should be noted that this did not put any severe constraints in achieving the solutions
as in all the ca^es these were obtained within 16 seconds of CPU time.
For eighteen elements, the final design vector, equality and .inequality constraints
along with the optimal value of the objective function are given in Table 3.2. The order
of magnitude of the equality constraints is small enough to be considered equal to zero
for all practical purposes. All the inequality constraints are negative showing that these
are strictly satisfied.
Figs. 3.2(a) and 3.2(b) show the variation of the absolute value of the objective
function with penalty parameter and the number of function evaluation respectively.
From the figures it can be seen that the objective function attains a constant value when
penalty parameter reaches a value of IC'^and the corresponding number of function
evaluations is 73334. The steady nature of the objective function indicates a convergent
solution.
Objective function
Fig 3.2 Variation of objective function with (a) Penalty parameter and
(b)Number of function evaluations
As lower -bound analysis involves the generation of statically admissible stress field,
it is of interest to study the state of stress in the soil medium corresponding to the
optimal solution at the limiting state. The nearness of the state of stress at nodal point
to the limiting state is judged by the stress strength ratio, defined as
[(cTj + ax) sin <I> + 2C cos <f>Y
where, ax and a^ are the normal stresses on the plane through a nodal point in x and
z direction respectively, t^x is the shear stress acting on the zx plane through a nodal
point, C is the cohesion and (j> is the angle of internal friction of the soil.
The complete stress field along with the stress strength ratio is shown in Table 3.3.
It can be seen from the table that the obtained stress field is excellent as the stress-
strength ratio at different nodal points for all the elements are very close to unity thus
signifying the limiting equilibrium state.
A5
Table 3.3
Stress Field and Stress-Strength Ratios at the
Nodal Points for eighteen elements
Element
No.
Nodal Point
No.
^zx
Stress Strength
ratio
1
1
15.4204
71.2003
0.0000
0.9506
1
2
15.4204
75.2055
0.0000
0.9999
1
3
15.4205
75.2055
0.0000
0.9999
2
1
15.3154
71.1834
0.0420
0.9562
2
3
15.4205
75.2055
0.0000
0.9999
2
4
15.5256
75.1887
0.0000
0.9940
3
1
14.4538
70.6321
0.7312
0.9990
3
4
15.5256
75.1887
0.0000
0.9940
3
5
16.1955
74.8230
0.0000
0.9535
4 '
1
15.9511
73.1624
1.2151
0.9479
4
5
16.1955
74.8231
0.0000
0.9535
4
6
16.2371
75.6737
0.0000
0.9568
5
1
12.9328
61.0891
4.8215
0.9999
5
6
14.0811
66.6897
4.4919
0.9962
5
7
13.3008
60.0142
6.6437
0.9962
6
1
11.6547
40.6404
9.9336
0.9996
6
7
12.4963
47.1428
9.8615
0.9999
6
8
12.3571
38.0969
11.0273
0.9960
7
1
11.5977
29.4715
10.7314
0.9999
7
8
12.3380
34.3410
11.2956
0.9999
7
9
12.3189
26.8534
11.1751
0.9961
8
1
11.5690
23.0060
10.3004
0.9837
8
9
12.3107
25.0217
11.0529
0.9977
8
10
12.2371
23.1742
10.7112
0.9802
9
1
11.1911
12.2577
8.2851
0.9991
9
10
11.9035
13.6866
8.9322
0.9969
9
11
11.6153
11.4019
8.1628
0.9999
10
1
10.1463
6.5225
5.8372
0.9962
10
11
11.0244
8.1584
6.7784
0.9991
10
12
10.2865
5.7454
5.4532
0.9944
. . . contd. on next page
^6
11
1
9.2873
4.2026
4.4225
0.9999
11
12
9.9722
4.9095
4.9378
0.9999
11
13
9.3521
3.7494
4.1065
0.9976
12
1
8.2344
2.5191
3.0935
0.9949
12
13
8.8583
2.9570
3.4809
0.9999
12
14
8.2551
2.2618
2.8439
0.9928
13
1
6.8313
1.1160
1.6904
0.9999
13
14
7.4100
1.4167
1.9988
0.9995
13
15
6.7402
0.9606
1.4557
0.9967
14
1
5.6190
0.4341
0.7811
0.9971
14
15
6.0652
0.5809
0.9494
0.9998
14
16
5.5468
0.3693
0.6270
0.9971
15
1
4.5396
0.0606
0.1462
0.9997
15
16
4.9581
0.1656
0.2807
0.9998
15
17
4.4038
0.0251
0.0086
0.9999
16
1
4.2531
0.0014
0.0160
0.9929
16
17
4.3484
0.0136
0.0165
0.9995
16
18
4.3044
0.0059
0.0243
0.9981
17
1
4.2787
0.0037
0.0237
0.9955
17
18
4.3218
0.0075
0.0190
0.9997
17
19
4.2870
0.0000
1
0.0252
0.9998
18
1
4.1285
0.0000
0.0000
0.9728
18
19
4.2873
0.0000
0.0252
0.9998
18
20
4.2884
0.0000
0.0000
0.9999
To study the extensibility of the stress field the original mesh for eighteen elements
was extended as shown in Fig. 3.1. The new solution for this mesh was found to
be 61.98 differing by only 0.70% from the previous one. Since the deviation is very
marginal and insignificant for all practical purposes, the stress field may be considered
to be extensible. Thus the obtained solution may be considered to be a true solution.
Since the analysis for the extended mesh is a general one, such study has not been
repeated for meshes with other number of elements.
In Table 3.4, the obtained solutions are compared with the exact value and that
from Bottero-Sloan’s approach.
Table 3.4 Comparison of Bearing Capacity Solutions
Exact Value of V, = 64.20
PRESENT SOLUTION
SLOAN’S SOLUTION
No. of
^0
% Diff. from
No. of
P*
% Diff from
Elements
Exact Soln.
Elements
Exact Soln.
6
47.84
25.48
6
35.68
44.42
12
5.45
12
53.58
16.54
18
62.42
2.77
12
24
59.69
24
62.65
2.41
48
61.35
4.43
36
62.56
2.55
OO
61.11
4.81
48
61.91
3.56
18*
61.98
3.45
^ T? = No. of Sides in linearized polygon
* Results for Extended mesh
Table 3.4 shows that the best solution from the present study differs by only 2.41%
from the exact solution on the safer side whereas that of Sloan’s differs from the same
by 4.43%. The corresponding number of elements required to get the solution are 24
and 12 respectively. Sloan made a piece-wise linear approximation of the non-linear
no-yield condition whereas for the Lysmer-Basudhar approach there was no necessity
for such an approximation of the no-yield condition. However, with 12 elements the
48
method predicted a value which is marginally smaller than that predicted by Sloan.
Taking eighteen elements instead of twelve elements a better solution (62.42) closer to
the exact value (64.20) was obtained by Lysmer Basudhar approach. When the number
of elements are twenty four there is a marginal increase in the Ng factor, beyond which
further increase in number of elements infact reduces this factor. As such there is no
need for consideration of number of elements more than twenty four for the estimation
of the Ng factor. However, even eighteen elements would give excellent results.
From the table a direct comparison of the computational efficiency of the two ap-
proaches could not be made as the number of function evaluations to achieve the final
optimal solution for twelve elements with Sloan’s approach is not available. However,
a qualitative and quantitative estimate can be made by comparing the final results
obtained by these two approaches.
With twelve elements and forty eight sides of the linearized polygon Sloan’s approach
presented a better solution (61.35) than the one (60.7) obtained with the same number
of elements and by using Lysmer-Basudhar approach ; the difference between these
solutions is only 1.06%. But, when only twenty four sides of the linearized polygon
are used Sloan’s approach predicts a value (59.69) less than the value (60.7) obtained
by the Lysmer-Basudhar approach differing by 1.69%. So by substantially increasing
the number of sides better values can be predicted by the Sloan’s approach than the
Lysmer-Basudhar approach. However, the table also shows that just by increasing the
number of elements from twelve to eighteen one predicts a better value (62.42) bj' the
49
Lysmcr-Basudhar ai)proach in contrast to the solution (61.35) obtained by using Sloan’s
approach with twelve ekuncnts and fortj’ eight sides; the relative clifTercnce between these
two solutions is 1.74% but the first one is closer to the exact one. It can be seen that
Sloan’s approach is strongly dependent on the number of sides of the linearized polygon
simulating the no-yield condition. But Lysmer-Basudhar approach does not have any
such drawbacks and as such, even with twelve elements a value of 60.7 could be obtained
whereas Sloan’s approach could not predict comparable values with less than twenty
four sides of the linearized polygon.
3.3 Footing on Cohesive Soil
Another simple problem of surface strip footing on saturated fine grained soils under
undrained condition has been chosen and is presented as follows.
3.3.1 The Problem
Fig. 3.3 shows a smooth strip surface footing resting on cohesive soil with Su = 1.0
kPa. The objective is to determine the bearing capacity factor Nc for this footing.
Fig 3.3 Strip footing on cohesive soil
50
3.3.2 The Objective Function
The objective function is — (ai 2 + o' 2 i). Bearing capacity q/ is equal to half of the
absolute value of the objective function.
3.3.3 The Boundary Conditions
The boundary conditions for the meshes shown in Fig. 3.3 are
<^23 = <^32 = 0 and
Ti2 = T2i = T 23 = T32 = Tis = Tgl = 0
3.3.4 Results and Discussions
The adopted mesh geometry is shown in Fig. 3.3. The Bearing capacity factor Nc
has been initially obtained for the mesh geometry (Fig. 3.3) with a equal to 22.5°(6
elements). Subsequently the same was estimated with reduced value of a(lO°) in the
radial shear zone and thus increasing the number of elements to 11. The mesh pattern
is so chosen as to enable a direct comparison with the values obtained from the present
approach with that of Munro-Chuang approach. For these values of a(22.5° and 10°)
the corresponding lower bound values of Nc using the present approach are 4.98 and 4.96
in comparison to the upper bound solutions (Munro-Chuang approach) 5.18 and 5.15
respectively. The lower bound solutions obtained by Lysmer (1970), Aral and Tagyo
(1985) and Sloan (1988) are 5.03, 5.04 and 5.08 respectively. The value 5.04 obtained by
Arai and Tagyo is with twelve elements; he obtained a value of 4.67 when the elements
wei’e increased to twenty four. Sloan and Lysmer obtained the coiiesponding values
51
with eight and six elements respectively.Arai and Tagyo stated that the higher value
of Nc with lesser number of elements is probably due to the over evaluation of the
footing pressure caused by the rough discretization of stress field. Such studies were
not conducted by Lysmer and Sloan whereas with both six and eleven elements, a
convergent solution has been obtained from the presented approach.
3.4 Conclusions
The following generalized conclusions, based on the presented results and discus-
sions, can be drawn:
• The stress field obtained by using the Lysmer-Basudhar approach has been found
to be extensible and, as such, the predicted solution is a true lower bound.
• The obtained bearing capacity factors using Lysmer-Basudhar approach are closer
to the exact solution than the same predicted by Bottero-Sloan’s approach. The
absolute errors of these two solutions from that of the exact solution (64.20) are
1.55 and 2.85 respectively and the corresponding relative errors are 2.41% and
4.43%. The adopted approach presented the best result very close to the exact
solution for twenty four elements, but, further increase in the number of elements
resulted in a marginal perturbation in the solution.
• Bottero-Sloan’s approach of finding the lower bound solution is strongly depen-
dent on the number of sides of the linearized polygon simulating the no-yield
condition whereas Lysmer-Basudhar approach does not suffer from anj^ such
52
drawbacks a5 it incorporates the non-linear no-yield constraints directly in the
analysis.
• The bearing capacity factor Nc obtained by the present method for saturated
fine grained soil under undrained condition is in close agreement with the values
reported in literature. The percentage difference of the present solution from that
of Lysmer (1970), Arai and Tagyo (1985) Sloan (1988) and Chuang (1992) are 1,
1.96, 1.19 and 2.9 respectively.
• Contrary to the general practice, retention of Powell’s conjugate direction algo-
rithm for unconstrained minimization in the Lysmer-Basudhar approach has been
found to be prudent from the fact that it could handle large number of design
variables without any problem.
CHAPTER 4
LOWER BOUND BEARING CAPACITY OF SURFACE
STRIP FOOTINGS ON TWO LAYERED SOIL DEPOSITS
4.1 Introduction
The bearing capacity of homogeneous soils has been the subject of extensive study.
But, in general, footings are to be located on natural stratified soil deposits exhibiting
varying strength characteristics. A very common kind of such soil deposits is a soil layer
of finite thickness overlying a thick stratum of another soil. The underlying stratum
may either be a bed rock or another soil layer possessing different strength properties.
Chapter 1 of the thesis gives a brief account of the available bearing capacity solutions
for both homogeneous and stratified deposits. It is noticed that, as compared to the
availability of bearing capacity solutions for homogeneous soil systems, the literature
to predict the same for stratified deposits is less. The methods employed are based on
the theory of linoiting equihbrium (eg. Button, 1953; Mandel and Salencon 1972), finite
element analysis (eg. Desai and Reese, 1970; Azam et ah, 1991), experimental studies
(eg. Tcheng, 1957; Brown and Meyerhof, 1969) lower bound limit analysis using the
method of characteristics (Krishnamurthy, 1972; Reddy et al.,1989 and 1990), finite
elements and non-linear programming (eg. Aral and Tagyo, 1985) and upper bound
limit analysis (eg. Purushottamaraj et al.,1974; Reddy and Rao, 1983). As such, it is
evident that apart from the application of the method of characteristics the only attempt
to use other generalized methods to predict the lower bound limit load for such layered
deposits has been made by Arai and Tagyo (1985). As the number of layers increases it
54
is very likely that the method of characteristics would be more difficult to use. As such,
discrete elements and optimization based techniques being more flexible and general
will be more appropriate for such problems. However, as already mentioned, only one
such attempt has been made for a two layered soil deposits. So it is necessary to develop
or extend other similar methods to such problems to asses their capability and validate
the theoretical predictive model by comparing the obtained solutions with experimental
observations as well as with other solutions reported in literature. With this in view
one such method namely Lysmer-Basudhar approach (Lysmer, 1970; Basudhar, 1976;
Basudhar et-al. 1979 and 1981) has been chosen and applied to study its suitability in
solving such a problem and the same is presented as follows.
4.2 The Problem
Fig. 4.1 shows a strip footing of width B lying on the surface of a soil layer of
thickness H having shear strength parameters Ci , 4>i overlying another soil stratum
having shear strength parameters C 2 and ^ 2 - The objective is to determine the bearing
capacity of this footing for the different cases shown in Fig. 4.2.
h- BH
IlftIfH
Layer 1 C-) ^ 01
Layer 2 C2i02
ii
H
V
Fig 4.1 Details of Surface Strip Footing on Two Layered Soil Deposit
56
The base of the foundation was considered to be rough for cases 1,2 and 3 (Brown
and Meyerhof, 1969) and that for cases 4 (Aral and Tagyo, 1985) and 5 (Krishnamurthy,
1972) to be smooth. The zone under consideration is divided into a number of elements
and the nodal points, elements and element sides are numbered in some arbitrary man-
ner. The meshes with marked nodal point numbers, element numbers and element sides,
used for analyses are shown in Figs. 4.2. It should be noted that the same element
sides and nodes at the layer interface have been marked differently to take care of the
different soil properties of the upper and lower layer. This also helps in taking care of
the possibility of discontinuity of the stresses at the interface.
4.3 The Objective Function
The objective function is — (<Ti 2 + cr 2 i) for all cases shown in Figs. 4.2. Bearing
capacity qj is equal to half of the absolute value of the objective function value.
4.4 The Boundary Conditions
The boundary conditions for the meshes shown in Figs. 4.2 are
<T 23 = <732 = 0 (for all cases)
for rough base(cases 1,2 and 3)
t ‘23 = T 32 = Tib = Tsi = 79,18 = Tjs.g = 0 (for cases 1 and 2)
(for case 3)
723 = 732 = 7i 9 = Tgi — 0
7i 2 = Cl -b (7 i 2 tan 6 and 721 = Ci -b <721 tan S
( for cases 1,2 and 3)
57
for smooth base(cases 4 and 5)
7 'i 2 = ^21 = 'r23 = T32 = Tjg = Tg] = Tg jg = Tjg g = 0
4.5 Results and Discussions
The lower bound bearing capacity solutions were obtained on CONVEX C-220
computer system. For saturated clay under undrained condition (<^i = <^2 = 0)
the computed value of the modified bearing capacity factor, A^cm, are compared with
the experimental observations reported by Brown and Meyerhof (1969). For case
l(C' 2 /C'i = 0.4 and ff/R = 0.25), the obtained modified bearing capacity factor Ncm
is 2.45 which is 2.45% on the higher side of the experimental value 2.4 . Similarly
for case 2(C2/Ci = 0A,H/B = 0.5) and case diCi/Ci = 0.2, HjB = 1.0), the ob-
tained values are 2,85 and 2.64 which, compared to the experimental values 2.8 and
2.5, are on the higher side by 1.7% and 4.3% respectively. Such a small difference in
the values can be neglected as these are well within experimental errors. So it can be
inferred that the Lysmer- Basudhar predictive model is excellent and the solutions can
be treated as a true indication of the critical load for footing with rough base. For case
4 (C 2 /Ci = 0.2,HfB = 2/3) a better lower bound estimation of Ncm (1-72) is obtained
from the present approach which is 1.71.69 reported by Arai and Tagyo (1985). However,
the observed difference is more in case of footing with smooth base resting on a general
cohesive frictional — layered soil deposit where foi C 2 /C 1 — 0.4, <^ 2 /*/*! — 1.25(with
= 15°) and H/B = 0.4, the obtained modified bearing capacity factor Ncm is 5.95
58
which is 12.5% on the lower side of the value 6.8 predicted by Krishnamurthy (1972) us-
ing method of characteristics. For better appreciation these solutions are also presented
in a tabular form (Table 4.1).
Table 4.1
Comparison of Bearing Capacity Factors for Footings on Two Layered Soil
Deposits
Case
H/B
C 2 IC 1
4>2/4>i
Brown and
Meyerhof*
(1969)
Krishna-
murthy
(1972)
Aral and
Tagyo
(1985)
Present
Study
% Diff.
1
0.25
0.4
MM
2.4
—
2.45
2.0
2
0.50
0.4
2.8
-
—
2.85
1.7
3
1.00
0.2
2.5
—
—
2.64
4.3
4
2/3
0.2
^9
—
—
1.69
1.72
1.7
5
0.40
0.4
1.25
6.8
—
5.95
12.5
^Experimental Results
For case 3, the final design vector, equality and inequality constraints along with the
objective function value at the optimum obtained starting from an arbitrarily chosen de-
sign vector are given in Table 4.2. The order of magnitude of the equality constraints is
small enough to be considered equal to zero for all practical purposes. All the inequality
constraints are negative showing that these are strictly satisfied.
Figs. 4.3(a) and 4.3(b) show the variation of the objective function with penalty
parameter and the number of function evaluations respectively. From these figures
it can be observed that the objective function attains a constant value when penalty
59
Table 4.2
Final ^^ign Vec^r, Sigma Vector, Constraints and
jective Function Value for eighteen elements
(D) Vector
0.3430
0.5778
0.4416
0.0004
0.9771
0.5457
1.6303
1.6393
2.6093
2.6888
3.6068
3.0197
0.5399
0.4392
0.5098
0.3323
0.5720
0.3899
0.6498
0.6676
1.6109
0.7321
0.7868
2.7966
0.6779
0.8356
0.7141
0.2915
0.3642
0.6012
0.4370
0.3415
0.3881
0.3314
0.3631
0.1684
0.2886
0.4746
0.2840
2.4529
0.7378
0.1927
0.6447
1.5192
0.9003
2.4930
Sigma Vector
2.4525
0.9909
1.5192
1.6109
0.3430
0.2396
0.0000
0.7378
0.5802
0.4746
0.4416
0.2346
0.3712
0.2429
0.1923
2.4930
2.7966
0.0000
0.0000
0.5346
1.6409
0.6474
1.8233
0.8702
1.1372
1.2213
0.6498
0.6096
0.2915
0.1230
0.1139
0.0111
0.0712
0.0000
0.5778
0.3333
0.0004
0.9771
0.5457
1.6303
1.6393
2.6093
2.6888
3.6068
3.0197
0.5399
0.4392
0.5098
0.3323
0.5720
0.3899
0.5176
0.6676
0.7870
0.7321
0.7868
0.8814
0.6779
0.8356
0.7141
0.5119
0.3642
0.6012
0.4370
0.3415
0.3881
0.3314
0.3631
0.1684
0.2886
0.3166
0.2840
0.2411
0.4321
0.1927
0.6447
0.4069
0.9003
0.3771
Interface Shear Equality Constraints
-8.3446E-07
5.9604E-08
1.8284E-07
2.9802E-07
-1.7816E-07
O.OOOOE 00
2.4026E-08
O.OOOOE 00
2.6195E-07
-5.9604E-08
-2.9538E-08
-1.7027E-07
8.0049E-08
-5.9604E-08
2.1535E-08
1.4901E-07
3.5816E-08
-8.9407E-08
-8.4756E-09
5.9604E-08
-1.1561E-09
1.4901E-08
3.7090E-08
-8.9407E-08
-4.4703E-08
-5.9604E-08
. . . contd. on next page
60
Boundary Shear Equality Constraints
2.9802E-08
-1.0658E-14
2.3841E-07
-1.7881E-07 -1.7816E-07
O.OOOOE 00
Constraints at Layer Interface
-2.0000E-01
-2.0000E-01
-2.0000E-01
-2.0000E-01 -2.0000E-01
-2.0000E-01
-2.0000E-01
-2.0000E-01
-2.0000E^01
-2.0000E-01
Non-linear No-yield Constraints (Inequality)
-3.2143E-01
-2.0989E 00
-5.501 8E-02
-3.1826E-03 -2.9897E-03
-7.0309E-03
-2.7489E 00
-7.9393E-04
-5.5069E-03
-4.7166E-03
-7.5731E-03 -1.8689E-01
-7.3600E-01
-4.0919E-01
-2.4981E-02
-8.7400E-01
-3.8828E 00
-8.1801E-03 -4.0000E00
-3.9952E 00
-3.6624E 00
-1.3827E-01
-9.4530E-02
-1.2023E-01
-1.2584E-01 -1.1760E-01
-7.9354E-02
-1.2325E-01
-1.5131E-01
-9.4758E-02
-1.4884E-01
-1.5452E-01 -9.0286E-02
-9.1440E-02
-7.0488E-02
-1.5812E-01
-1.5921E-01
-1.3938E-01
-1.3638E-01 -4.2824E-02
-8.7521E-02
-1.432lE-0i
-3.7421E-02
-6.6932E-02
-1.2258E-01
No-tension
Constraints
(Inequality)
-2.4525E 00
-9.9092E-01
-1.5192E OO
-1.6109E0O -3.4301E-01
-2.3962E-01
-7.3784E-01
-5.8021E-01
-4.7469E-01
-4.4160E-01
-2.3465E-01 -3.7122E-01
-2.4290E-01
-1.9233E-01
-2.4930E 00
-2.7966E 00
-5.3463E-01
-1.6409E 00 -6.4745E-01
-1.8233E 00
-8.7026E-01
-1.1372E 00
-1.2213E 00
-6.4985E-01
-6.0964E-01 -2.9154E-01
-1.2309E-01
-1.1395E-01
-1.1131E-02
-7.1299E-02
8.9923E-08
-5.7780E-01 -3.3331E-01
-4.1735E-04
-9.7715E-01
-5.4577E-01
-1.6303E 00
-1.6393E OO
-2.6093E 00 -2.6888E00
-3.6068E 00
-3.0197E 00
-5.3999E-01
-4.3923E-01
-5.0986E-01
-3.3239E-01 -5.7203E-01
-3.8997E-01
-5.1762E-01
-6.6767E-01
-7.8705E-01
-7.3212E-01
-7.8689E-01 -8.8145E-01
-6.7790E-01
-8.3562E-01
-7.1410E-01
-5.1193E-01
-3.6427E-01
-6.0122E-01 -4.3700E-01
-3.4150E-01
-3.8811E-01
-3.3148E-01
-3.6314E-01
-1.6847E-01
-2.8864E-01 -3.1660E-01
-2.8408E-01
-2.4117E-01
-4.321 lE-01
-1.9272E-01
-6.4473E-01
-4.0697E-01 -9.0035E-01
-3.7714E-01
Optimal function value = 5.29
61
parameter reaches a value of 10 ^ and the corresponding number of function evaluations
is 46859. This steady nature of the objective function indicates a convergent solution.
Fig 4.3 Variation of objective function with (a) Penalty parameter and
(b)Number of function evaluations
The complete stress field along with the stress-strength ratio is shown in Table 4.3.
It can be observed from the table that some of the nodal points are veiy close to limiting
state. Similar stress fields are obtained for other cases, but for the sake of space and
brevity they have not been presented herein.
Studies on extensibility have been undertaken for a typical case (case 3). The absolute
value of the objective function for the original mesh [Fig. 4.2(c)] is 5.28. The nodal
points 16,17,18 and 19 are extended downwards as shown in Fig. 4.4(a) and the value
obtained for this case is 5.29 showing no remarkable increase from the previous value:
then nodal points 3,4 and 15 arc also extended along with the nodal points 16,17,18
and 19 ,Fig. 4.4(b), but the value 5.29 still remain unchanged.
62
Table 4.3
Stress Field and Stress-Strength Ratios at the
Nodal Points for case 3
Element
No.
Nodal Point
No.
Cx
(Tz
Tzx
Stress Strength
ratio
1
1
0.5346
2.4525
0.0000
0.9196
1
9
1.6409
3.0197
0.0000
0.4752
1
8
1.6409
3.6068
0.1417
0 9862
2
2
0.8139
2.7966
0.1282
0.9992
2
1
0.4983
2.4930
0.0674
0.9992
2
8
1.3999
2.7664
0.7289
0.9982
3
2
1.0876
1.5192
0.5159
0.3127
3
8
0.8460
2.6888
0.3883
0.9998 •
3
7
0.6116
2.6093
0.0303
0.9986
4
2
1.4644
1.6109
0 9967
0.9988
4
7
0.8646
1.6393
0.9209
0.9981
4
6
0.9006
1.6303
0.9056
0.9532
5 •
2
1.5525
0.3430
0.6710
0.8159
5
6
1.2832
0.5457
0.8727
0.8977
5
5
1.2053
0.9771
0.9903
0.9937
6
2
1.5439
0.2396
0.5968
0.7814
6
5
0.3425
0.0004
0.0050
0.0292
6
4
1.8005
0.3333
0.6780
0.9979
7
3
0.0000
0.0000
0.0000
0.0000
7
2
0.0687
0.0000
0.0000
0.0011
7
4
0.5778
0.0000
0.0302
0.0843
8
14
0.6444
0.7870
0 0186
0.1357
8
16
0.9640
0.7235
0.0435
0.4091
8
15
0.6544
0.7321
0.0918
0.2485
9
14
0.7611
0.5802
0.0188
0.2134
9
17
0.5829
0.3771
0.0031
0.2649
9
16
0.6722
0.9003
0.0846
0.5040
10
13
0.3642
0.5176
0.0575
0.2296
10
17
0.5585
0.5530
0.0465
0.0542
10
14
0.5648
0.6676
0.1169
0.4077
11
13
0.3898
0.4416
0.0046
0.0697
11
18
0.3365
0 4069
0 0114
0 0342
11
17
0.4462
0 6447
0.0870
0.4357
12
13
0.3222
0.3899
0 1264
0 4284
12
12
0.2909
0 5720
0 0512
0.5594
12
18
0 3447
0.3033
0.0062
0.0117
13
12
0.3788
0 3712
0.0134
0.0049
13
19
0.2526
01927
0.0652
0.1288
13
18
0.2846
0.4321
0.0216
0.1475
14
11
0.2886
0.5098
0 1306
0.7323
14
19
0.3166
0.2543
0.1309
0.4529
14
12
0.2889
0.3323
0.0610
0.1049
15
11
0.2886
0.4392
0.1580
0.7661
15
10
0.2367
0.5399
0.0166
0.5816
15
19
0 3166
0.1564
0.0542
0.2338
64
Finally two more elements 16 and 17 are added to the original mesh pattern of Fig.
4.2(c), the modified pattern is shown in Fig 4.4(c). The objective function value ob-
tained for this case is 5.26 which is very marginally different from the values 5.28 and
5.29. To study the extensibility of this mesh pattern, the meshes have been extended as
shown in Fig. 4.4(c) and the obtained objective function value is 5.28, again showing no
appreciable difference from the value obtained with the unextended mesh pattern indi-
cating that the stress field is extensible. Thus the obtained solution may be considered
to be a true lower bound solution.
4.6 Conclusions
The following generalized conclusions, based on the presented results and discussions,
can be drawn;
• Lysmer-Basudhar approach using discrete elements and non-linear programming
can be used quite efficiently and reliably for predicting the lower bound bearing
capacity of surface strip footings on a two layered soil deposit.
• The obtained lower bound bearing capacity factors for rough footings on two
layered fine graind soil deposits under undrained condition ((^„ = 0) are marginally
higher (1.7 - 4.3%) than the available experimental values. The difference is well
within permissible experimental error.
• For a general C - 4> layered soil deposit, the obtained value of Ncm. for smooth
footing is 12.5% on the lower side of the value obtained using method of charac-
teristics.
• Lysmer-Basudhar approach gives a better estimation of lower bound bearing ca-
pacity factor than that from Aral and Tagyo approach.
CHAPTER 5
GENERALIZED CONCLUSIONS AND SCOPE FOR
FUTURE STUDIES
5.1 Conclusions
The generalized conclusions that are drawn based on the studies reported in chapters
3 and 4 are presented as follows:
• i) All the methods of isolating the optimal solution of bearing capacity problems
based on linear programming (Lysmer, 1970; . Sloan, 1988; Chuang, 1992) and
non-linear programming (Basudhar, 1976; Arai and Tagyo, 1985) predicts values
which are in close agreement for homogeneous soil deposits and the variations in
the results are no more than 2%-3%.
• ii) Bottero-Sloan approach is considerably influenced by the number of sides of the
linearized polygon simulating the no-yield condition whereas Lysmer-Basudhar
does not suffer from any such drawbacks as it incorporates the non-linear no-yield
constraints directly in the analysis.
• iii) Lysmer-Basudhar approach has been found to be quite efficient and reliable
for predicting the lower bound bearing capacity of surface strip footing on a two
layered soil deposit and predicts better solution than that by Arai and Tagyo.
The predicted values using this method closely agree with the experimental ob-
servations. However, these solutions differs from that of method of characteristics
solution by about 12.5%.
66
5.2 Scope for Future Studies
• i) As choice of the mesh pattern is very important and at present is mostly guided
by intution, previous experience and experimental observations, more studies are
needed to provide guidelines for proper discretization of the medium.
• ii) As the adopted method of analysis does not provide any information regarding
the displacement, appropriate constitutive relationship for the soil be included in
the model and a mixed formulation be made. This would result in obtaining an
exact solution.
• iii) Extension of Lysmer— Basudhar approach to reinforced soils simulating the
presence of the soil and the reinforcing elements separately than treating the
whole soil-reinforced composite mass as an equivalent horhogeneous anisotropic
material.
• iv) Design charts based on lower bound solutions for unconventional problems
in soil mechanics like bearing capacity of footings located above voids may be
prepared.
APPENDIX
PROOF THAT IT IS SUFFICIENT TO SATISFY THE NO- YIELD
CONDITION AT THE CORNERS OF THE ELEMENTS
It has been proved by Lysmer (1993) that, if the no-yield condition is satisfied at
the corners of an element, no points within the element will be above yield. The same
is reproduced here as follows.
The no-yield condition is:
[<^Z - ^xf + < [(cr, -f- CTj.) sin •+ 26’ cos 4>f (1)
or, it can be written as
A^ + Ei^ < ( 2 )
where A, B and C are generalized stresses :
A = a^ — Ox
B — Ir^x
C = (cTj -t- (Tx) sin 4> + 2C cos d* (3)
These stresses vary linearly over the aar-plane since and t^x have this property.
Also C > 0, since a no-tension stress field is assumed. Let us now consider two points,
0 and 1, at which the linear stress field. For these points we have by equation (2),
pointO Tq^Aq -f Bl) = Cq Co > 0
point! t\{A\ -h B^) = Ci > 0
(4)
68
where tq > 1 and ri > 1 are known constants.
The geneialized stresses at points on a line between points 0 and 1 vary linearly.
Thus
A[t) — (1 — t)Ao + tAi
= + (5)
C{t) = (1 — t)CQ + tCi
where 0 < i < is a parameter which has the values 0 and 1 at points 0 and 1 respectively.
Substuting equation (4) into the expression for C{t) we get,
C(f) = (1 - t)rQy/Al + B^ + + Bl (6)
Thus the right hand side of the equation (2) is:
C^t) = (1 - tfrl + Bl) + firliAl + Bf)
+2t(l - i)ror, ^(AIaBI)(AIaBI) (7)
which for points between points 0 and point 1 satisfy
C\t)> {l-t)^{Al + Bl)+tHAl + Bl)
+2t{l-t)^{Al + Bl){Al + Bl] ( 8 )
The left hand side of (2) is, by (5),
A^ii) + B^(t) = (1 - tf{Al + ^o) + + -^i) + " 0(>lo-4i + BoBi) (9)
69
'll the Ta.'\ge 0 < t < 1, we have 2t(l — t) > 0. Thus by comparison between (8) and
')), we c».i3clude that the no-yield condition
A\t) + B\t)<C\t) (10)
is satisfied in the range 0 < t < 1, provided it is true that
Ao>li + BoBr < yJ{Al -f Bl){Al + B?) (11)
I'his is incleed so, which can be seen from the following:
{AqBi — BqAi)^ > 0 (self evident)
{AoB^y -h (5oAa)2 > 2 Ao5i^oAi
Adding {AqAi^ + {BoBiY to both sides, we get
{Al + Bl){Al + B^,) > {AoA, + BoB^Y
y/{Al + B^){Al + B^) > |AoAi + BoBi\
which c€)nfirins that (11) and thus (10) is satisfied. But this means that, in a linear
stress Jifddj all points on a line between two no-yielding points will be non-yielding
Thij) theorem guarantees that, if the no-yield condition is satisfied at the coiners of
an elenrmt, no points within the element will be above yield.
REFERENCES
Arai, K. and Tagyo, K. (1985), “Limit Analysis of Geotechnical Problems by Applying
Lower Bound Theorem”, Soils and Foundations, 25, No. 4, 37-48.
Assadi, A. and Sloan, S. W. (1991), “Undrained Stability of A Shallow Square Tunnel”,
Journal of Geotechnical Engineering, ASCE, 117, No. 8, 1152-1173.
Azam, G., Hsieh, C.W. and Wang, M.C. (1991), “Performance of Strip Footing on
Stratified Deposit With YoidF , Journal of Geotechnical Engineering, ASCE, 117, No.
5, 753-772.
Baker, R. and Frydman, S. (1983), “Upper Bound Limit Analysis of Soil With Non-
linear Failure Criterion”, Soils and Foundations, 23, No. 4, 34-42.
Basudhar, P. K. (1976), “Some Applications of Mathematical Progia- mming Tech-
niques to Stability Problems in Geotechnical Engineering” , Ph.D. thesis, Indian Insti-
tute of Technology, Kanpur, India.
Basudhar, P. K., Madhav, M. R. and Valsangkar, A. J. (1979), “Optimal Lower Bound
of Passive Earth Pressure Using Finite Elements and Nonlinear Programming’, Inter-
national Journal for Numerical and Analytical Methods in Geomechanics, 3, 367-379.
Basudhar, P. K., Madhav, M. R. and Valsangkar, A. J. (1981), “Sequential Uncon-
strained Minimization in the Optimal Lower Bound Bearing Capacity Analysis , Indian
Geotechnical Journal, 11, 42-55.
Bans, R. L. and Wang, M. C.-(1983), “Bearing Capacity of Strip Footing Above Void”,
Journal of Geotechnical Engineering, ASCE, 109, No. 1, 1-14.
Belytschko, T. and Hodge, P. G. (1970), “Plane Stress Limit Analysis By Finite Ele-
ments”, Journal of Engg. Mech. Div,^ ASCE, 96, 931-944.
Bottero, A., Negre, R., Pastor, J. and Turgeman, S. (1980), “Finite Element Method
and Limit Analysis Theory for Soil Mechanics Problems”, Comput. Methods Appl.
Mech. Engg., 22, 131-149.
Brown, J. D. and Meyerhof, G. G. (1969), “Experimental Study of Bearing Capacity
in Layered Clays” Proceedings, Seventh International Conf Soil Mech. and Found.
Engg., Mexico City, 2, 45-51.
Button, S. J. (1953), “The Bearing Capacity of Footings on a Layer Cohesive
Subsoil”, Proc. Third Int. Conf Soil Mech. and Found. Engg., Zurich, 1, 33--335.
Casciaro, R. and Cascini, L. (1982), “A Mixed Formulation and Mixed Finite Elements
for Limit Analysis”, International Journal for Numerical and Analytical Methods in
Geomtchanics^ 18, 211-243.
Chen, W. F. and Davidson, H. L. (1973), “Bearing Capacity Determ- ination Ly Limit
Analysis”, Journal of the Soil Mechanics and Foundations Division, ASCE, 99, No.
SM 6, 433-449.
71
Chen , W. F. and Scawthorn, C. R. (1970), “Limit Analysis and Limit Equilibrium
Solutions in Soil Mechanics”, Soils and Foundations, 10, No. 3, 13-49.
Davis, E. H. (1968), “Theories of Plasticity and the Failure of Soil Masses”, Soil Me-
chanics - Selected Topics, Ed. 1. K. Lee, Chapter 6, American Elsevier, New York,
341-380.
Davis, E. H. and Booker, J. R. (1973), “The Effect of Increasing Strength With Depth
on the Bearing Capacity of Clays”, Geoitchnique, 23, No. 4, 551-563.
De Borst, R. and Vermeer, P. A. (1984], “Possibilities and Limitations of Finite Ele-
ments for Limit Analysis”, Geotechnique, 34, No. 2, 199-210.
Desai, C. S. and Reese, L. C. (1970), “Analysis of Circular Footings on Layered Soils”
Proceedings ASCE, Journal of the Soil Mechanics and Foundations Division, 96, No.
SM4, 1289- 1310.
Drucker, D. C. (1953), “Limit Analysis of Two and Three Dimensional Soil Mechanics
Problem”, Journal of the Mechanics and Physics of Solids, London, 1, 217-226.
Drucker, D.-C., Greenberg, H. J. ajid Prager, W. (1952), “Extended Limit Design
Theorems for Continuous Media”, Quarterly Journal of Applied Mathematics, 9, 381-
389.
Drucker, D. C. and Prager, W. (1952), “Soil Mechanics and Plastic Analysis or Limit
Design”, Quarterly Journal of Applied Mathematics, 10, 157-165.
Finn, W. D. L. (1967), “Application of Limit Plasticity in Soil Mechanics”, Journal of
the Soil Mechanics and Foundations Division, ASCE, 93, No. SM 5, 101-120.
Fox, R. L. (1971), “Optimization Methods for Engineering Design”, Addison- Wesley,
Reading, Mass.
Gioda, G.and Donato, 0. D. (1979), “Elastic-Plastic Analysis of Geotechnical Problems
by Mathematical Programming”, International Journal for Numerical and Analytical
Methods in Geomechanics, 3, 381-401.
Graham, J. (1968), “Plane Plastic Failure in Cohesion less Soils”, Geotechnique, 18,
301-316.
Hanna, A. M. (1981), “Experimental Study on Footings in Layered Soil”, Journal of
Geotechnical Engineering Division, ASCE, 107, No. GT8,1113 — 1127.
Hanna, A. M. (1982), “Bearing Capacity of Foundation on Weak Sand Layer Overlying
a Strong Deposit”, Canadian Geotechnical Journal, 19, No. 3, 392-396.
Kavlie, D. and Moe, J. (1971), “Automated Design of Frame Structures”, Journal of
Structural Division, ASCE, 97, ST1,33 — 61.
Krishnamurthy, S. (1972), “Limiting Equilibrium Solutions to a Class of Stability Prob-
lems in Soil Mechanics”, Ph.D. thesis, Indian Institute of Technology, Kanpur, India.
72
Kiisakabe, 0., Kimura, T. and Yamaguchi, H. (1981), “Bearing Capacity of Slopes
Under Strip Loads on the Top Surface”, Soils and Foundations^ 21, No. 4, 29-40.
Lysmer, J. (1970), “Limit Analysis of Plane Problems in Soil Mechanics”, Journal of
the Soil Mechanics and Foundations Division, ASCE, 96,SM4, 1311 — 1334.
Lysmer, J. (1993), “Proof that it is Sufficient to Satisfy the No-yield Condition at the
Corners of the Elements”, Personal Communication.
Mandel, J. and Salencon, J. (1969), “Force Portante d’un Sol Sur Une Assise Rigide”,
Proceedings Seventh International Conf. Soil Mech. and Found. Engg., Mexico City,
2, 157-164.
Mandel, J. and Salencon, J. (1972), “Force Portante d’um Sol Sur Une Assise Rigide
(Etude Theorique), Geotechnique 22, No. 1, 79-93.
Meyerhof, G. G. (1951), “The Ultimate Bearing Capacity of Foundations”, Geotech-
nique, 2, No. 4, 301-332.
Meyerhof, G. G. (1974), “Ultimate Bearing Capacity of Footings on Sand Layer Over-
lying Clay”, Canadian Geotechnical Journal, 11, No. 2, 223-229.
Mizuno, E. and Chen, W. F. (1983), “Cap Models for Clay Strata to Footing Loads ,
Comp, and Sirs., 17, No.4, 511-528.
Powell, M. J. D. (1964), “An Efficient Method for Finding the Minimum of A Function
of Several Variables Without Calculating Derivatives”, Computer J., 7, No. 4, 303-307.
Purushothamaraj, P., Ramiah, B. K. and Rao, K. N. V. (1974), Bearing Capacity
of Strip Footings in Two Layered Cohesive - Friction Soils”, Canadian Geotechmque
Journal, 11, No. 1, 32-45.
Rao, S. S. (1984), “Optimization Theory and Application”, Wiley- Eastern Limited.
Reddy, A. S., Dutt, H. H. and Jagannath, S. V. (1989), “Bearing Capacity of Circular
Footing in Two Layered Soil”, Indian Geotechnical Journal, 19, No. 2, 167-180.
Reddy, A. S-, Jagannath, S. V. and Dutt, H. H. (1990), “Bearing Capacity of Footings
on Two Layered Soil”, Indian Geotechnical Journal, 20, No. 3, 161-174.
Reddy, A. S. and Rao, K. N. V. (1983), “Bearing Capacity of Strip Footing in Two
Layer C - <i> Soils Exhibiting Anisotropy and Non-homogenity m Cohesion , Indian
Geotechnical Journal, 13(4), 187-210.
Sabzevari, A. and Ghahramani, A. (1972),- “The Limit Analysis of Bearing Capacity
and Earth Pressure Problems in Nonhomogeneous Soils”, Soils and Foundations, 12,
No. 3, 33-48.
Salencon, J. (1977), “Application of the theory of plasticity m Soil Mechanics , John
Wiley and Sons. inc. New York.
73
Satyanarayana, B. and Garg, R. K. (1980), “Bearing Capacity of Footings on Layered
C — 4> Soil”, Journal of Geotechnical Engineering Division, ASCE, 106, No. GT7,
Proc. Paper 15578, 819-824.
Singh, D. N. (1992), Lower Bound Solutions of Some Stability Problems in Geotech-
nical Engineering , Ph.D. thesis, Indian Institute of Technology, Kanpur, India.
Singh, D. N. and Basudhar, P. K. (1992), “A Note on the Optimal Lower Bound Pullout
Capacity of Inclined Strip Anchor in Sand”, Canadian Geotechnical Journal, 29 , No.
5, 870-873.
Singh, D. N. and Basudhar, P. K. (1993), “Determination of the Optimal Lower Bound
Bearing Capacity of Reinforced Soil Retainig Walls by Using Finite Elements and Non-
linear Programming”, Geoiexiiles and Geomemhranes, 12 , In Press.
Singh, D. N. and Basudhar, P. K. (1993), “A Note on Vertical Cuts in Homogeneous
Soils”, to appear in Canadian Geotechnical Journal, August Issue.
Siva Reddy, A. and Srinivasan, R. J. (1967), “Bearing Capacity of Footings on Layered
Clays”, Proceedings ASCE, Journal of Soil Mechanics and Foundations Division, 93,
No. SM2,83 - 99.
Sloan, S. W. (1988), “Lower Bound Limit Analysis Using Finite Elements and Linear
Programming”, International Journal for Numerical and Analytical Methods in Geome-
chanics, 12, 61-77.
Sloan, S. W. (1989), “Upper Bound Limit Analysis Using Finite Elements and Linear
Programming”, International Journal for Numerical and Analytical Methods in Geome-
chanics, 13, 263-282.
Sloan, S. W., Assadi, A. and Purushothaman, N. (1990), “Undrained Stability of A
Trapdoor”, Geotechnique, 40, No. 1,45-62.
Sokolovsky, V. V. (1960), “Statics of Soil Media”, Butiermorth, London.
Sokolovsky, V. V. (1965), “Statics of Granular Media”, Pergamon press, Oxford.
Tamura, T., Kobayashi, S. and Sumi, T. (1984), “Limit Analysis of Soil Structures by
Rigid Plastic Finite Element Methods”, Soils and Foundations, 24, No. 1, 34-42.
Tamura, T., Kobayashi, S. and Sumi, T. (1987), “Rigid-Plastic Finite Element Method
for Frictional Materials” , Soils and Foundations, 27 , No. 3, 1-12.
Tcheng, Y. (1957), “Foundations Superficielles tn milieu stratifie”. Fourth Int. Conf.
Soil Mech. and Found. Engg., London, 1,449-452.
Terzaghi, K. (1943), “Theoretical Soil Mechanics”, John Wiley and Sons., Inc., New
York, N.Y.
Vesic, A. (1975), “Bearing Capacity of Shallow Foundations”, Foundation F- gir ffC'-.g
Handbook. Eds. Winterkorn and H.Y.Fang, Van Nostrand Reinhold Company Inc.,
New York, N.Y.
74
Yamaguchi, H. (1963), “Practical Formula for Bearing Value for Two Layered ground”,
Proc. Second Asian Regional Conf. on Soil Mech. and Found. Engg., Japan, 1, 176-
180.
Yokowo, Y., Yamagata, K. and Nagaoka, H. (1968), “Bearing Capacity of a Continuous
Footing Set in Two Layered Ground”, Soils and Foundations, Japan, 8. No. 3, 1-31.
Yong, R. N. and Mohamed, A. M. 0. (1991), “Nonlinear Stress Analysis of Muskeg via
Finite Element”, Canadian Geotechnical Journal, 28, No. 4, 613-629.
Live Music
Archive
Librivox
Free Audio
Metropolitan Museum
Cleveland
Museum of Art
Internet
Arcade
Console Living Room
Open Library
American
Libraries
TV News
Understanding
9/11